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10.12688/f1000research.172934.1

Research Article

Articles

Indeterminacy of Boolean Ring

[version 1; peer review: 2 approved with reservations]

Ahmed

Yousif A.

Writing – Review & Editing https://orcid.org/0009-0000-5308-0356 a 1 Mohammed Abed

Majid

Writing – Review & Editing b 1 1Mathematic, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

a you24u2002@uoanbar.edu.iq b majid_math@uoanbar.edu.iq

No competing interests were disclosed.

6 2 2026

2026

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19 1 2026

2026

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Background

A neutrosophic ring represents an algebraic generalization of the classical ring structure by introducing an indeterminacy element I , enabling the modeling of truth, falsity, and indeterminacy simultaneously, as established within Smarandache’s neutrosophic framework. In contrast, a Boolean ring is a commutative algebraic structure in which every element is idempotent ( αI ) ² = αI reflecting the logical principles of Boolean algebras and possessing characteristic two Combining these concepts, the neutrosophic Boolean ring extends the Boolean ring by embedding neutrosophic logic parameters—truth (T), indeterminacy (I), and falsity (F)—into its elements and operations. This hybrid structure allows for the representation of algebraic uncertainty and incomplete information while preserving Boolean idempotent properties, thus providing a flexible framework for studying systems with uncertain or partially defined information in algebraic and logical contexts

Methods

The research defines the Indeterminacy ring R I = { α + βI : α , β ∈ R } and explores its algebraic properties through examples from integers, rationals, and reals. It then formulates the Indeterminacy Boolean Ring (B-Ring) characterized by idempotency ( αI ) ² = αI , and establishes several theorems proving its core algebraic features.

Results

Findings reveal that Indeterminacy B-Rings are commutative and have characteristic two, ensuring 2 αI = 0 . Each maximal Indeterminacy ideal is also prime, and these rings are semisimple and reduced, containing no nonzero nilpotent elements. Furthermore, any Indeterminacy B-Ring can be represented as a direct product of copies of Z ₂ I , known as the Indeterminacy Boolean field. The quotient rings preserve Boolean and Indeterminacy properties, confirming their structural consistency.

Conclusions

The study successfully extends Boolean ring theory to the Indeterminacy domain, establishing a strong algebraic foundation for modeling uncertainty. Indeterminacy B-Rings maintain the essential Boolean properties of idempotency and commutativity while incorporating indeterminate behavior through I ² = I . These results open new perspectives for future applications in neutrosophic logic, fuzzy systems, and abstract algebra dealing with indeterminate information.

Indeterminacy ring; Prime ideal; Maximal ideal; Idempotent; Indeterminacy Boolean ring; semi simple; homomorphism; Indeterminacy Boolean field

No grants supported the work

The author(s) declared that no grants were involved in supporting this work.

1. Introduction

Fuzzy theory is one important of many branches in mathematic. Many authors have investigated indeterminacy-based algebraic structures. In particular, Smarandache introduced the general framework of (T, I, F)- Indeterminacy structures and explored their algebraic properties. ¹ In Agboola, ² several fundamental results in lattice theory were developed, forming a structural basis for algebraic systems involving order relations and ideal theory. These foundations can be extended to study the behavior of neutrosophic and indeterminacy-based lattices, where it is proved that in Indeterminacy B-Rings every maximal ideal is also a prime ideal (Chalapathi and Madhavi). ³ More results related to idempotent elements have been presented in Al-Hamido, A., ⁴ The property of idempotency plays a central role in Boolean and Indeterminacy B-Rings. Ali and Smarandache ⁵ presented a comprehensive survey of neutrosophic and indeterminacy-based algebraic systems, outlining the general framework of Indeterminacy algebra. Later, Chalapathi and Madhavi further developed the structural aspects of Indeterminacy B-Rings. Also, in, ^{15–
17} some information about fuzzy ideal and some definitions in Indeterminacy theory. In, ^{6,
7} the authors presented an integrated framework of Indeterminacy set. The existence of a multiplicative identity e=I in (R ∪ I) guarantees structural stability and allows generalization to broader algebraic contexts. ³ Additionally, illustrative examples were presented to highlight cases where idempotency holds or fails, alongside remarks connecting algebraic logic with the structural properties of Indeterminacy B-Rings. ⁸ Thus, this research focuses on Indeterminacy groups, maximal ideals, prime ideals, idempotent, and especially Indeterminacy B-Rings, presenting new results and extending the existing ones to enrich the field of Indeterminacy algebra These insights pave the way for further applications in fuzzy mathematics and the theory of ideals. Finally, in the context of algebraic logic, the connections between Indeterminacy rings, Indeterminacy groups have been investigated to highlight the structure of Indeterminacy evaluation rings, their commutativity, and their role in lattice theory.

2. Methods

Note: Throughout this article, the notation “ R I ” will be used in place of R I Definition 2.1.

¹¹ Consider S as a non-empty set. An Indeterminacy set A I on X is known as: A I = { ( α , T A ( α ) , I A ( α ) , F A ( α ) ) : α ∈ S } ,

were

T A ( α ) ⊆ [ 0 , 1 ] , (Truth-membership),

I A ( α ) ⊆ [ 0 , 1 ] , (Indeterminacy membership),

F A ( α ) ⊆ [ 0 , 1 ] (Falsity membership).

of an element x in the set A I .

Remark 2.2.

The fundamental feature of an Indeterminacy set is that it generalizes classical, fuzzy, and intuitionistic fuzzy sets by explicitly incorporating indeterminacy.

Example 2.3.

Let S = { α , β } . Define An Indeterminacy set A as: A = { ( α , 0.7 , 0.2 , 0.1 ) , ( β , 0.4 , 0.3 , 0.6 ) } .

Here, element a belongs to A with truth degree 0.7, indeterminacy 0.2, and falsity 0.1. Definition 2.4.

¹² For ( G , ⋅ ) , We have that it is a group. An Indeterminacy group is defined as: ( G ∪ I ) = { α + βI : α , β ∈ G } , where I is an indeterminate element with I ² = I . The group operations are extended naturally from G .

Example 2.5.

For G = ( Z 3 , + ) the additive group of integers modulo 3. So, Indeterminacy group is: ( G ∪ I ) = { 0 , 1 , 2 , 0 + I , 1 + I , 2 + I } .

For example, ( 1 + I ) + ( 2 + I ) = ( 1 + 2 ) + ( I + I ) = 0 + I .

Definition 2.6.

⁸ For R to be a ring. R I = { α + βI : α , β ∈ R } is referred to as Indeterminacy ring generated by R and I .

Remark 2.7.

The indeterminate element I satisfies the condition I ² = I , which is essential in defining Indeterminacy rings.

Definition 2.8.

³ Let R be a ring. The Indeterminacy ring R I is a ring generated by R and I .

Remark 2.9.

The angle bracket notation R I is sometimes used to emphasize the closure under ring operations.

Example 2.10.

⁶ We denote by Z the ring of integers, Z I = { α + βI : α , β ∈ Z } .

This is a ring termed the Indeterminacy ring of integers. Also, Z ⊊ Z I .

Remark 2.11.

The enlargement from Z to Z I highlights how Indeterminacy extensions generalize classical rings.

Example 2.12.

⁴ We denote by Q the ring of rationales. Q I = { λ + βI : λ , β ∈ Q } . This is the Indeterminacy ring of rationales.

Definition 2.13.

¹³ We denote by R a ring. A subset J I ⊆ R I is referred to as Indeterminacy ideal of the Indeterminacy ring R I if for all r ∈ R and j ∈ J , we have: rj ∈ J , jr ∈ J , and ( j + I ) r , r ( j + I ) ∈ J I .

Remark 2.14.

The presence of the indeterminate I ensures that classical ideals extend naturally into the Indeterminacy framework.

Example 2.15.

Let R = Z 6 and J = { 0 , 2 , 4 } . Then J is an ideal in R . The Indeterminacy ideal is: J I = { 0 , 2 , 4 , 0 + I , 2 + I , 4 + I } ⊆ Z 6 I .

Definition 2.16.

¹⁴ An Indeterminacy field is An Indeterminacy algebraic structure ( F I , + , ⋅ ) where F is a classical field and I is the Indeterminacy indeterminate with I ² = I . It satisfies all field axioms extended with the Indeterminacy component.

Example 2.17.

Let F = Q , the field of rational numbers. Then the Indeterminacy field is: F I = { α + βI : α , β ∈ Q } .

For example, ( 1 + I ) ⋅ ( 2 + I ) = 2 + 3 I + I ² = 2 + 3 I + I = 2 + 4 I .

Remark 2.18.

Although Q is a field, Q I is not a field since I ² = I and I has no multiplicative inverse. Still, it is sometimes loosely designated the Indeterminacy field of rationales.

Example 2.19.

⁵ Let R be fixed as the ring of real numbers. R I = { λ + sI : λ , s ∈ R } . This is the Indeterminacy ring of real’s.

Remark 2.20.

Similarly to the rationals, it is only a ring and not a true field, but in literature it is sometimes termed the Indeterminacy field of real’s.

Example 2.21.

⁸ C I = { z + wI : z , w ∈ C . This is the Indeterminacy ring of complex numbers.

Remark 2.22.

Even though C is algebraically closed and a field, its Indeterminacy extension C I is not a field because of the special Indeterminacy element I.

Definition 2.23.

Let R I be An Indeterminacy ring. It is said to be comm. if ∀ αI , βI ∈ R I , ( αI ) ( βI ) = ( βI ) ( αI ) . If in addition ∃ I ∈ R I ∋ 1 · λ = λ · 1 = λ , ∀ λ ∈ R I , then R I is termed a comm. Indeterminacy ring with unity.

Remarks 2.24.

1. ⁷ Unity here generalizes the multiplicative identity of the base ring R .

2. ³ Again, although Q is a field, Q I is only a ring since I lack an inverse.

3. It is not a field, but in many Indeterminacy studies it is referred to as the Indeterminacy field of complex numbers.

3. Results

In this part, we present some new results about important ring in abstract algebra and more properties have been studied. But before that we, need some definitions and examples in order to study this ring in Indeterminacy theory. Definition 3.1.

¹⁰ Any ring R is termed Boolean if α 2 = α , ∀ α ∈ R ).

Definition 3.2.

For a B-Ring R , We say R I is Indeterminacy B-Ring if for every element in R I αI ∈ R I , then ( αI ) 2 = α 2 I 2 = αI .

Example 3.3.

An Indeterminacy ring ( Z 2 I , + 2 , . 2 ) is An Indeterminacy B-Ring.

Example 3.4.

An Indeterminacy ring ( P ( X ) , ∆ , ∩ ) , Where P ( X ) = { AI : AI ⊆ X I } is An Indeterminacy B-Ring Since ( P ( X ) , ∆ , ∩ ) is An Indeterminacy ring with identity and ∀ AI ∈ P ( X ) ⟹ A 2 I 2 = AI ∩ AI = AI .

Example 3.5.

An Indeterminacy ring ( Z 3 I , + 3 , . 3 ) is not Indeterminacy B-Ring. Since ∃ 2 I ¯ ∈ Z 3 I is not idempotent element, s.t 2 2 I 2 = 2 I . 3 2 I = I ≠ 2 I .

Example 3.6.

For ( R I , + , ∙ ) comm. Indeterminacy ring with unity

R I = { ψ ; ψ : X I → Z 2 I } and ∀ αI ∈ X I we have: ( ψ + ϕ ) ( αI ) = ψ ( αI ) + 2 ϕ ( αI ) ( ψϕ ) ( αI ) = ψ ( αI ) ∙ 2 ϕ ( αI )

So, ( R I , + , ∙ ) be a comm. Indeterminacy ring with unity. It achieves the following and ∀ ψ ∈ R I either ψ ( αI ) = 0 or ψ ( αI ) = 1 . Then ψ 2 = ψ . If ψ ( αI ) = 0 .

Then, ψ ( αI ) 2 = ψ ( αI ) ∙ 2 ψ ( αI ) = 0 ∙ 2 0 = 0

Or ψ ( αI ) = 1

Hence, ψ ( αI ) 2 = ψ ( αI ) ∙ 2 ψ ( αI ) = I ∙ 2 I = I

And ψ 2 = ψ

Thus, ( R I , + , ∙ ) Indeterminacy B-Ring.

Theorem 3.7.

Consider R I as An Indeterminacy B-Ring. Then ( − αI ) = αI , ∀ αI ∈ R I .

Proof:

We prove that if R I is a ring, so ∀ αI , βI ∈ R I , ( − αI ) ( − βI ) = αβI and ( − αI ) 2 = αI , ( − αI ) ( − αI ) = ( αI ) 2 . From the definition of An Indeterminacy B-Ring, ( αI ) 2 = α 2 I 2 = αI ⟹ ( αI ) 2 = αI … . . ∗ .

Thus, ( − αI ) 2 = ( − αI ) . But ( − αI ) 2 = ( αI ) 2 . So, ( αI ) 2 = ( − αI ) .

from * we get αI = ( − αI ) and this required.

Theorem 3.8.

Every Indeterminacy B-Ring ( R I , + , . ) with the characteristic 2 has the property 2 αI = 0 I .

Proof:

Let αI ∈ R I , and since R I is An Indeterminacy B-ring

Therefore, αI + αI ∈ R I

So,

αI + αI = ( αI + αI ) 2 ( R I is a B-Ring)

Hence, αI + αI = α 2 I 2 + 2 α 2 I 2 + α 2 I 2

Therefore, αI + αI = αI + 2 αI + αI ( αI ∈ R I ⟹ α 2 I 2 = αI )

Then, 0 = 2 αI ⟹ 2 αI = 0 , ∀ αI ∈ R I

Thus, h ( R ) = 2 .

Theorem 3.9.

Consider R I as An Indeterminacy B-Ring. Then R I is comm. under ( · ) .

Proof:

Assume that I , βI ∈ R I . We need to show that αβI = βαI s . t ( αI ) ( βI ) = αβ I 2 = αβI ⟹ ( αI + βI ) = ( αI + βI ) 2

(since αI , βI ∈ R I and is Indeterminacy B-Ring) ⟹ ( αI + βI ) = ( αI + βI ) ( αI + βI ) ⟹ ( αI + βI ) = ( αI ) 2 + αIβI + βIαI + ( βI ) 2

But R I An Indeterminacy B-Ring we have ( αI ) 2 = αI and ( βI ) 2 = βI ⟹ ( αI + βI ) = αI + αIβI + βIαI + βI ⟹ 0 = αIβI + βIαI ⟹ αIβI = − αIβI but αI = − αI from Theorem 3.7

Hence αIβI = αIβI as required.

Theorem 3.10.

If R I is An Indeterminacy ring with identity. Then, every Indeterminacy maximal ideal is Indeterminacy prime ideal.

Proof:

Assume that R I Indeterminacy ring with identity and ( P I , + , ∙ ) An Indeterminacy maximal ideal in R I . To verify that ( P I , + , ∙ ) is Indeterminacy prime ideal. Let αI , βI ∈ R I , ∋ αI ∙ βI ∈ P I and let αI ∉ P I

( P I , + , ∙ ) An Indeterminacy maximal ideal in R I , and αI ∉ P I .

Then PI + ( αI ) = R I . Hence I ∈ R I ⟹ I ∈ PI + ( αI ) ⟹ I = δI + λIαI , λI ∈ R I , δI ∈ P I and I 2 = I , I = δI + λαI } ∗ βI βI ∙ I = δI ∗ βI + λIαI ∗ βI , βI = δβI + λαβI . Therefore δI ∈ P I , βI ∈ R I ⟹ δβI ∈ P I . Hence λI ∈ R I , αI ∙ βI ∈ P I ⟹ λαβI ∈ P I . Then δβI + λαβI ∈ P I . So βI ∈ P I . Thus ( P I , + , ∙ ) is Indeterminacy prime ideal .

Theorem 3.11.

Let R I be An Indeterminacy B-Ring and S I be An Indeterminacy ideal in that Indeterminacy ring. Then P is Indeterminacy prime ideal iff it is Indeterminacy maximal ideal.

Proof:

⟹ Let S I Indeterminacy prime ideal. We need to prove that S I is Indeterminacy maximal ideal. Take an P I Indeterminacy ideal in R I . s.t ( S ⊂ P ⊆ R) S I ⊂ P I ⊆ R I , To show that P I = R I .

Hence S I ⊂ P I ⟹ ∃ αI ∈ P I , αI ∉ S I . Then αI ∈ P I ⟹ αI ∈ R I since P I ⊆ R I .

But R I is Indeterminacy B-Ring, then ( αI ) 2 = αI

αI ( I − αI ) = 0 ∈ S I . So αI ∉ S I , S I Indeterminacy prime ideal.

Therefore, ( I − αI ) ∈ S I , Hence S I ⊂ P I , And so, ( I − αI ) ∈ P I , . Also, ( I − αI ) + αI ∈ P I .

Then, I ∈ P I , So P I = R I . Thus, S I is Indeterminacy maximal ideal. ⟸

Let S I be An Indeterminacy maximal ideal. To prove S I is An Indeterminacy prime ideal. Hence R I is An Indeterminacy B-Ring. Also, R I abelian Indeterminacy ring with identity and by Theorem 3.11 (if R I Indeterminacy ring with identity. Then, every Indeterminacy maximal ideal is Indeterminacy prime ideal). Thus S I Indeterminacy prime ideal.

Remark 3.12.

(1) Let ( R I , + , ∙ ) be a comm. Indeterminacy ring with ( 1 I ) , The S I ⊆ R I such that a)

S I = ( αI ∈ R I | ( αI ) 2 = αI ) ,

αI + βI = αI + βI − 2 αβI , αI ∗ βI = αβI , ∀ αI , βI ∈ R I .

Then,

( S I , + , ∙ ) is An Indeterminacy B-Ring. S I = ( αI ∈ R I | ( αI ) 2 = αI ) , ∀ αI , βI ∈ R I αI + βI = αI + βI − 2 αβI , αI ∗ βI = αβI .

(2) Multiplication: If αI , βI ∈ S I , then ( αβI ) ² = ( αI ) ² ( βI ) ² = αβI , so αβI ∈ S I .

(3) Addition: If αI , βI ∈ S I , then αI + βI = αI + βI − 2 αβI .

Also, ( αI + βI − 2 αβI ) ² = ( αI + βI − 2 αβI ) ( αI + βI − 2 αβI ) = ( αI + βI ) ( αI + βI ) − 2 αβI ( αI + βI ) − 2 αβI ( αI + βI ) + ( 2 αβI ) ( 2 αβI ) = ( αI ) 2 + 2 αβI + ( βI ) 2 − 2 ( αI ) 2 βI − 2 αI ( βI ) 2 − 2 ( αI ) 2 βI − 2 αI ( βI ) 2 + 4 ( αI ) 2 ( βI ) 2 = αI + 2 αβI + βI − 2 αβI − 2 αβI − 2 αβI − 2 αβI + 4 αβI = αI + βI − 2 αβI , so αI + βI ∈ S I , also ( S I , + ) is an abelian group.

(4) Comm.: αI + βI = βI + αI

Associativity: holds by expansion.

Identity of ( S I , + ) is 0 s.t I + 0 = 0 + αI = αI .

Inverses in ( S I , + ) , αI + α − 1 I = 0 , so every element is its own inverse.

Thus ( S I , + ) is an abelian group.

(5) Distributive: For αI , βI , γI ∈ S I :

αI ( βI + γI ) = αβI + αγI − 2 αβγI = ( αβI ) + ( αγI ) . So distributivity holds.

(6) Boolean property: For every αI ∈ S I : each element is idempotent under multiplication, and ( S I , + , ∙ ) satisfies all ring axioms and every element is idempotent under multiplication. Hence ( S I , + , ∙ ) is a B-Ring.

Corollary 3.13.

If ( R I , + , . ) is An Indeterminacy ring with identity I . Then, ( R I / P I , + , . ) is also Indeterminacy ring with identity I + δI .

Proof:

Since R I is An Indeterminacy ring with identity I , Then + δI ∈ R I / P I . Hence I + δ I is an identity element of R I / P I with respect multiplication, Since ∀ αI + δI ∈ R I / P I . Hence ( αI + δ I ) . ( I + δ I ) = ( αI . I ) + pI = αI + δ I and ( I + δ I ) . ( αI + δ I ) = ( I . αI ) + δ I = ( αI + δ I ) . Therefore, R I / P I is Indeterminacy ring with identity element + δI .

Theorem 3.14.

For ( R I , + , . ) , An Indeterminacy B-Ring. Then, ( R I P I , + , . ) is also Indeterminacy B-Ring.

Theorem 3.15.

Consider ( R I , + , · ) as An Indeterminacy B-Ring. For any αI ≠ 0 , βI ≠ 0 , γI ≠ 0 ∈ R I , Then ( αI + βI ) ( βI + γI ) ( γI + αI ) = 0 .

Proof:

In An Indeterminacy B-Ring, every element is idempotent, that is ( αI ) ² = αI for all αI ∈ R I . ⁷ From idempotency, one derives that the ring has characteristic 2 . Indeed, ( αI + αI ) ² = αI + αI implies 2 αI = 0 , hence αI + αI = 0 for all αI ∈ R I . ⁹ Since ( αI ) ² = αI and ( βI ) ² = βI , we have αβI + βαI = 0 . In characteristic 2 , this simplifies to αβI = βαI . An Indeterminacy B-Ring is comm. So, ( αI + βI ) ² = ( αI ) ² + αβI + βαI + ( βI ) ² = αI + βI . ⁵ Multiplying this ( αI + βI ) ( βI + γI ) ( γI + αI ) , ∀ αI , βI , γI ∈ ( R ∪ I ) s . t αI , βI and γI non-zero element.

Now ( αI + βI ) ( βI + γI ) = αβI + αγI + ( βI ) 2 + βγI = αβI + αγI + βI + βγI ,

Now ( αI + βI ) ( βI + γI ) ( γI + αI ) = ( αβI + αγI + βI + βγI ) ( γI + αI ) = ( αβγI + αβγI ) + ( αβI + αβI ) + ( αγI + αγI ) + ( βγI + βγI ) = 2 αβγI + 2 αβI + 2 αγI + 2 βγI = 0 + 0 + 0 + 0 .

Then ( αI + βI ) ( βI + γI ) ( γI + αI ) = 0 . ⁶

Corollary 3.16.

Let ( P I , + , ∙ ) be a proper ideal in the Indeterminacy B-Ring ( R I , + , · ) . Then P I is maximal iff ( R I / P I , + , · ) ≅ ( Z 2 I , + 2 , · 2 ) .

Proof:

Since ( R I , + , · ) is an Indeterminacy B-Ring, the quotient Indeterminacy ring ( R I / P I , + , · ) is also Indeterminacy Boolean. Moreover, as R I is a comm. Indeterminacy ring with identity, R I / P I inherits these properties and remains a comm. Indeterminacy ring with identity.

For any element αI + P I ∈ R I / P I , we have: ( αI + P I ) ² = ( αI + P I ) ( αI + P I ) = αI ² + P I = αI + P I .

Hence, R I / P I is an Indeterminacy B-Ring. It is well known that an ideal P I is maximal in R I iff R I / P I is an Indeterminacy field.

Corollary 3.17.

Every Indeterminacy B-Ring ( R I , + , · ) is semisimple, that is, rαd ( R I ) = { 0 } .

Proof:

Assume that ( R I , + , · ) be an Indeterminacy B-Ring. We aim to prove that R I is semisimple, i.e., rαd ( R I ) = { 0 } . Let, for contradiction, that rαd ( R I ) ≠ { 0 } . Then there exists a nonzero element αI ∈ rαd ( R I ) .

From the auxiliary lemma, there exists an Indeterminacy ring homomorphism ψ : R I → Z 2 I such that ψ ( αI ) = I . Consequently, ker ( ψ ) is a proper ideal of R I , and hence there exists a maximal ideal P I in R I such that ker ( ψ ) ⊆ P I .

Since 1 I − αI ∈ ker ( ψ ) ⊆ P I and αI ∈ P I (because αI ∈ rαd ( R I ) = P I -maximal), we get 1 I = αI + ( 1 I − αI ) ∈ P I , which implies that P I = R I , a contradiction. Therefore, rαd ( R I ) = { 0 } , and R I is semisimple.

Theorem 3.18.

Every Indeterminacy B-Ring is isomorphic to a direct product of copies of Z 2 I . Formally, R I ≅ ∏ { i ∈ I } Z 2 I .

Proof:

Every Indeterminacy B-Ring can be viewed as an Indeterminacy ring of functions from some index set P I to Z 2 I . This representation arises because each element of R I corresponds to unique Indeterminacy boolean combination of projections onto Z 2 I .

Proposition 3.19.

Every ideal in an Indeterminacy B-Ring is Indeterminacy radical ideal.

Proof:

Assume that P I an ideal of R I , and suppose αI ∈ P I , meaning αⁿ ∈ P I for some n ≥ 1 . But in an Indeterminacy B-Ring, ( αI ) ⁿ = αI , so αI ∈ P I . Hence, P I = P I .

Theorem 3.20.

Every Indeterminacy B-Ring is reduced (contains no nonzero nilpotent elements) and therefore semi-simple.

Proof:

If αI ∈ rαd ( R I ) , then a is nilpotent. But in an Indeterminacy B-Ring, ( αI ) 2 = αI , implying αI = 0 or 1 I . Since 1 I ∉ rαd ( R I ) , it follows that αI = 0 . Thus, rαd ( R I ) = 0 , and R I is semisimple.

Proposition 3.21.

For each element αI in an Indeterminacy B-Ring R I , there exists an onto Indeterminacy ring homomorphism ψ : R I → Z 2 I such that ψ ( αI ) = 1 I .

This shows that Indeterminacy B-Ring possess many surjective homomorphisms to Z 2 I , allowing R I to decompose as a direct product of copies of Z 2 I .

Corollary 3.22.

Up to isomorphism, there exists only one Boolean field, namely Z 2 I .

Proposition 3.23.

An Indeterminacy B-Ring ( R I , + , · ) is an Indeterminacy field iff ( R I / P I , + , · ) ≅ ( Z 2 I , + 2 , · 2 ) .

Proof:

Assume that ( R I / P I , + , · ) is an Indeterminacy B-Field. For any αI ∈ R I , the following holds: αI = αI · 1 I = αI ( αI · ( αI ) − 1 ) = ( αI ) ² · ( αI ) − 1 = αI · ( αI ) − 1 = 1 I .

Thus, R I = { 0 , 1 I } , and consequently R I ≅ Z 2 I . Therefore, P I is maximal in R I iff R I / P I is an Indeterminacy field, which occurs precisely when R I / P I ≅ Z 2 I .

Conclusion

Boolean ring is one of the important rings in abstract algebra. All results and properties of Boolean ring are presented in Indeterminacy theory. We proved in ( Theorem 3.7) the relation between Indeterminacy Boolean ring and the property ( − αI = αI .). Also, we proved that, every Indeterminacy Boolean ring ( R I , + , . ) with the characteristic 2 has the property 2 αI = 0 I . On the other hand, if R I is An Indeterminacy ring with identity so, every Indeterminacy maximal ideal is Indeterminacy prime ideal. In Corollary 3.14, we say if ( R I , + , . ) is An Indeterminacy ring with identity I . Then, ( R I / P I , + , . ) is also Indeterminacy ring with identity I + δI . Finally, more results in this article have been presented.

Discussion

This paper does not include a discussion section.

Ethical considerations

This article does not involve human participants or animal subjects.

Data availability

No datasets were generated or analyzed during the current study.

Reporting guidelines

All relevant research and reporting guidelines were appropriately followed.

Acknowledgements

The author would like to thank the reviewers for their valuable and constructive comments that helped improve the quality of the article.

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10.5256/f1000research.190701.r473985

Reviewer response for version 1

Gokavarapu

Chandrasekhar

1 Referee 1Acharya Nagarjuna University, Nagarjuna Nagar, Andhra Pradesh, India

Competing interests: No competing interests were disclosed.

6 5 2026

2026

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve-with-reservations

Detailed Report

1. Presentation and Literature Review

The introduction successfully contextualizes the work within fuzzy theory and Smarandache’s neutrosophic framework. However, there is a distinct "organizational gap" between the literature review and the technical proofs.

Criticism: The transitions between the historical context and the new mathematical developments are abrupt.

Recommendation: Strengthen the narrative by explicitly stating how the cited literature necessitates the specific theorems presented in Section 3.

2. Technical Soundness and Methodology

While the mathematical logic is sound and the theorems are relevant, the Methods/Preliminary section (Section 2) suffers from a lack of cohesion.

Criticism: Definitions and examples often appear disconnected, making it difficult for the reader to grasp the logical progression.

Recommendation: Group related definitions (e.g., combining the formal definition of an Indeterminacy B-ring with its illustrative example) to improve clarity. Introductory paragraphs should be added to each subsection to explain the utility of each concept.

3. Language and Formal Aspects

The most significant barrier to the article's impact is its current editorial state.

Grammar & Punctuation: There are frequent missing commas and periods, and inconsistent capitalization (e.g., "An indeterminacy" vs. "an indeterminacy").

Mathematical Style: The text frequently embeds symbolic quantifiers (like $\forall$ or $\exists$) directly into prose sentences.

Recommendation: Replace symbolic quantifiers with words (e.g., "for all" or "there exists") when they appear in the middle of a sentence to improve readability. A comprehensive professional proofreading is essential.

4. Conclusions

The conclusions are logically derived from the results, but they remain somewhat brief.

Recommendation: Expand the conclusion to discuss specific potential applications in fuzzy systems or neutrosophic logic, providing a "roadmap" for future research.

Required Amendments for Scientific Soundness

To move this article from "Approved with Reservations" to full approval, the following points must be addressed:

Reorganize Section 2: Create a clearer logical flow by adding transitions and grouping interdependent definitions.

Standardize Notation: Ensure the ring notation is consistent throughout the manuscript, particularly the transition between $R_L$ and $R_I$.

Language Correction: Fix grammatical errors, standardize capitalization, and ensure punctuation is mathematically and linguistically correct.

Contextualize Definitions: Provide a brief explanation for why a specific definition (e.g., Indeterminacy Field) is being introduced and how it serves the subsequent theorems.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

ALGEBRA, SYMMETRY, NONLINEAR DYNAMICS

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

adeeb

yousif

mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

Competing interests: No competing interests were disclosed.

19 5 2026

Response to Reviewer 1

We sincerely thank the reviewers for their careful reading of the manuscript and for their constructive comments. We are pleased that the reviewers found the topic relevant and the mathematical contribution valuable. We have revised the manuscript carefully in response to all comments.

First, we strengthened the connection between the literature review and the technical developments by adding a transitional paragraph at the end of the Introduction. This paragraph now explains how the previous work on neutrosophic and indeterminacy-based algebraic structures motivates the study of Indeterminacy Boolean rings and prepares the reader for the theorems presented in Section 3.

Second, we reorganized and clarified the preliminary section. We added introductory paragraphs before the main groups of definitions in Section 2. These paragraphs explain the role of Indeterminacy sets, Indeterminacy groups, Indeterminacy rings, ideals, and field-based extensions in the development of the main results. We also improved the logical flow between definitions and examples, especially around the concepts needed for Indeterminacy Boolean rings.

Third, we revised the mathematical notation throughout the manuscript. In particular, the notation RIRI is now used consistently to denote the Indeterminacy extension of a ring RR. We also reviewed the notation related to ideals, quotient structures, and Boolean ring conditions to ensure consistency.

Fourth, we improved the language, grammar, punctuation, and mathematical style. Symbolic quantifiers such as ∀∀ and ∃∃ were replaced by words when they occurred inside prose sentences. We also corrected capitalization problems, including expressions such as “An Indeterminacy” when used in the middle of a sentence.

Finally, we expanded the Conclusion to emphasize the significance of the results, possible applications in neutrosophic logic and fuzzy systems, and future research directions such as homomorphisms, automorphisms, and modules over Indeterminacy Boolean rings.

We believe that these revisions have improved the clarity, coherence, and scientific presentation of the manuscript.

10.5256/f1000research.190701.r456955

Reviewer response for version 1

Higuera Rincon

Sebastian David

1 Referee Rubiano Suárez

Andrés Alejandro

1 Co-referee https://orcid.org/0009-0009-1633-8018 1Universidad Antonio Narino (Ringgold ID: 27967), Bogotá, Bogotá, Colombia

Competing interests: No competing interests were disclosed.

5 3 2026

2026

recommendation

approve-with-reservations

The submitted article addresses a relevant topic in its research area and presents an interesting and potentially valuable contribution to the literature. Below, I provide my evaluation regarding clarity of presentation, methodological soundness, and the connection between results and conclusions.

1. General Presentation and Literature Review

The introduction of the article is clear and well written. It explains the current state of the topic in the literature and clearly defines the problem that the paper aims to study. The objectives of the work are presented in a precise way, and the potential contribution of the article is understandable.

However, it would be helpful to strengthen the connection between the literature review and the technical developments that appear later in the paper. Adding clearer transitions could help readers better understand the importance of the main results from the beginning.

2. Methods and Preliminary Section

The Methods (or preliminary) section includes many definitions, examples, and remarks that seem necessary for the development of the paper. However, this section is not always clearly organized. The information sometimes appears disconnected, and it is difficult to see a clear logical structure.

Although the preliminary material is important, I suggest the following improvements:

Provide more explanation about the context of each definition or concept.

Explain more clearly why each concept is important in the existing literature.

Indicate how each definition or result will be used later in the Results section.

Add short introductory paragraphs or transitions to guide the reader through the section.

In addition, some definitions and examples could be grouped together to avoid repetition and improve clarity. For example, the definition of Indeterminacy B-ring could be presented in a more compact way, combining the formal definition and an illustrative example in the same place.

These changes would improve the clarity and coherence of the paper.

3. Results and Conclusions

The Results section is generally well developed. The theorems presented are relevant and meet the expectations created in the introduction. There is a good connection between the objectives of the paper and the results obtained.

The conclusions are consistent with the results. However, the final section could be strengthened by emphasizing more clearly the importance of the results, their possible applications, and potential directions for future research.

4. Writing and Formal Aspects

One of the main aspects that needs improvement is the writing quality. The article contains several grammatical and punctuation errors that should be corrected before indexing. In particular:

Some commas and final periods are missing.

The ring notation should be carefully revised for consistency.

There are incorrect expressions in English, such as writing “An indeterminacy” instead of “an indeterminacy”.

In several places, symbolic quantifiers are used in the middle of sentences. It would be better to write them in words (for example, “for all x” or “there exists an x such that…”) to improve readability and style.

A careful language revision by a native English speaker or a professional proofreader is strongly recommended.

5. Overall Evaluation

In conclusion, this is an excellent work from a mathematical and conceptual point of view. The results are relevant and aligned with the objectives stated in the introduction.

However, the paper needs important improvements in the organization of the preliminary section and especially in the writing and presentation of the material.

The following points must be addressed to ensure that the article is scientifically sound and clearly presented:

Reorganize and clarify the preliminary section.

Provide more context for the definitions and concepts introduced.

Carefully revise grammar, punctuation, and language.

Correct and standardize the mathematical notation.

After these revisions, the article would have strong potential to become a solid and well-presented contribution to the field.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

My research areas are commutative and noncommutative algebra of polynomial type, module theory, and category theory.

We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above.

adeeb

yousif

mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

Competing interests: Don't have any interests must be declared.

30 4 2026

First: We extend our sincere thanks to the esteemed evaluator for what he said regarding the paper being good and its results being original, as well as the originality of the examples and applications presented in it.

The following is the response to the comments kindly provided by the esteemed resident, and they have been carefully considered due to their importance.

1- The paper is well written and technically sound. To further improve its quality, the following minor revisions are recommended. After addressing these comments, the paper may be considered for acceptance for indexing.

Answer: We extend our sincere thanks to the esteemed reviewer for the positive evaluation of our work and for recognizing that the paper is well written and technically sound. We highly appreciate the valuable comments and suggestions, which have significantly helped us improve the quality, clarity, and presentation of the manuscript. All the suggested revisions have been carefully addressed.

2- Rewrite the introduction in a standard and structured form.

Answer: The introduction has been completely rewritten in a clear, standard, and well-structured manner. In particular, we improved the logical flow by strengthening the connection between the literature review and the main contributions of the paper. We also clarified the research gap and explicitly highlighted the objectives and significance of the work.

3- Rewrite the preliminary in a standard and structured form.

Answer: The preliminary section has been thoroughly reorganized and rewritten to ensure clarity and logical coherence. Definitions, remarks, and examples are now presented in a consistent and structured sequence. Additional explanations and transitions have been added to clarify the role of each concept and its relevance to the results presented later in the paper.

4- Provide more context and explanation for definitions and concepts.

Answer: Additional explanations have been incorporated throughout the preliminary section to provide proper context for each definition and concept. We have clarified their importance in the existing literature and explicitly indicated how they are used in the subsequent results.

5- Correct and standardize mathematical notation.

Answer: All mathematical notations have been carefully reviewed and standardized throughout the manuscript. In addition, symbolic expressions within sentences have been replaced with appropriate textual forms (e.g., “for all” instead of symbols) to improve readability and consistency.

Acknowledgments: We would like to express our sincere gratitude to the reviewer for the constructive feedback and insightful comments. The revisions have substantially improved the quality and presentation of the paper, and we hope that the manuscript is now suitable for publication.

adeeb

yousif

mathematics, University of Anbar, Ramadi, Al Anbar Governorate, Iraq

Competing interests: No competing interests were disclosed.

19 5 2026

Response to Reviewer 1

We believe that these revisions have improved the clarity, coherence, and scientific presentation of the manuscript.