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

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 ) } .

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 .

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

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

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

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 .

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

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

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

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 .

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 .

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.

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.

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.

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

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.

Let \(R_I\) be an Indeterminacy ring. The ring \(R_I\) is called commutative if, for all \(\alpha I,\beta I\in R_I\), we have

\[

(\alpha I)(\beta I)=(\beta I)(\alpha I).

\]

If there exists an element \(1_I\in R_I\) such that

\[

1_I\cdot \lambda I=\lambda I\cdot 1_I=\lambda I

\]

for every \(\lambda I\in R_I\), then \(R_I\) is called an Indeterminacy ring with identity.

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.

Let \(R_I\) be an Indeterminacy ring. Then \(R_I\) is called an Indeterminacy Boolean ring if

\[

x^2=x

\]

for every \(x\in R_I\). In particular, if \(x=\alpha I\), then the idempotent condition becomes

\[

(\alpha I)^2=\alpha I.

\]

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

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 .

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 .

clarify the role of idempotency in distinguishing Indeterminacy Boolean rings from general Indeterminacy rings. The ring \(\mathbb {Z}_{2I}\) satisfies the Boolean condition because each element is idempotent, whereas \(\mathbb {Z}_{3I}\) fails to be an Indeterminacy Boolean ring because it contains non-idempotent elements. These examples justify the need for the formal results that follow.

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.

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

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.

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

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 .

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

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.

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

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 .

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.

⟹ 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.

(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 ) ,

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.

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

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 .

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

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 .

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

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 ) .

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.

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

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.

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

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 .

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

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 .

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

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.

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 .

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

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

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 .