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Prime and Maximal Ideals of Pre A*-Algebra



A. Satyanarayana, J. Venkateswara Rao and U. Suryakumar
 
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ABSTRACT

In this study we formulate the definitions of an ideal, prime ideal, maximal ideal of Pre A*-algebra A and discuss certain examples. We prove important fundamental properties of Ideals. In particular we extend to prove that every ideal I of a Pre A*-algebra A is the intersection of all prime ideals of A containing I. We also show that every maximal ideal is necessarily prime, while the converse is true for special cases only.

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  How to cite this article:

A. Satyanarayana, J. Venkateswara Rao and U. Suryakumar, 2011. Prime and Maximal Ideals of Pre A*-Algebra. Trends in Applied Sciences Research, 6: 108-120.

DOI: 10.3923/tasr.2011.108.120

URL: https://scialert.net/abstract/?doi=tasr.2011.108.120
 
Received: April 25, 2010; Accepted: May 17, 2010; Published: July 22, 2010



INTRODUCTION

The study of lattice theory had been made by Birkhoff (1948). In a draft paper, the equational theory of disjoint alternatives by Manes (1989) introduced the concept of Ada (Algebra of disjoint alternatives) (A, ∧, ∨, (-)', (-)π, 0,1,2) which is however differ from the definition of the Ada of E.G. Manes (1993) later study Adas and the equational theory of if-then-else. While the Ada of the earlier draft seems to be based on extending the If-Then-Else concept more on the basis of Boolean Algebra and the later concept is based on C-algebra (A, ∧, ∨,') introduced by Guzman and Squier (1990).

Koteswara (1994) first introduced the concept A*-Algebra (A, ∧, ∨,*, (-),(-)π, 0,1,2) not only studied the equivalence with Ada, C-algebra, Ada’s connection with 3- Ring, Stone type representation but also introduced the concept of A*-clone. Venkateswara Rao and Koteswara Rao (2003) studied about Boolean algebras and A*-algebras and the methods of generating A*-algebras from Boolean algebras and vice-versa. Venkateswara Rao and Koteswara Rao (2008) studied about If-Then-Else structure over A*-algebra and also Venkateswara Rao and Koteswara Rao (2004) studied about Prime ideals of A*-algebra. Venkateswara Rao and Koteswara Rao (2005) obtained Cayley theorem for A*-Algebras. Venkateswara Rao (2000) introduced the concept of Pre A*-algebra (A, ∧, ∨,*, (-)) as the variety generated by the 3-element algebra A={0,1,2} which is an algebraic form of three valued conditional logic. It was proved that the only sub directly irreducible Pre A*-algebra are either A or two element Boolean algebra B={0,1}. Venkateswara Rao et al. (2009) generated Pre A*-algebras from Boolean algebras and defined congruence relation and Ternary operation on Venkateswara Rao and Srinivasa Rao (2009) defined a partial ordering on a Pre A*-algebra A and the properties of A as a poset are studied.

In this study we formulate the definitions of an ideal, prime ideal, maximal ideal of Pre A*-algebra A and discuss certain examples. We also show that for any ideals I and J of a Pre A*-algebra A, the ideal (I: J) is precisely equal to the intersection of all prime ideals containing J and not containing I. We also show that every maximal ideal is necessarily prime, while the converse is true for special cases only.

PRELIMINARIES

Definition 1: Jocobson (1994) Boolean algebra is an algebra (B, ∨, ∧, (-)' 0,1) with two binary operations, one unary operation (called complementation) and two nullary operations which satisfies:

(B, ∨, ∧) is a distributive lattice
x ∧ 0 = 0, x ∨ 1 = 1
x ∧ x' = 0, x ∨ x' =1

  We can prove that x" = x,(x ∨ y)' = x' ∧ y', (x ∧ y)' = x' ∨ y', for all x, y ∈B.
  Here, we concentrate on the algebraic structure of Pre A*-algebra and state some results which will be used in the later text.

Definition 2: Venkateswara Rao (2000): An algebra (A, ∧, ∨, (-)~) where A is non-empty set with 1, ∧, ∨ are binary operations and (-) is a unary operation satisfying:

x~~ = x, ∀ x∈A
x ∧ x = x, ∀ x∈A
x ∧ y = y ∧ x, ∀ x, y∈A
(x ∧ y)~ = x~ ∨ y~, ∀ x, y ∈A
x ∧ (y ∧ z) = (x ∧ y) ∧ z, ∀x, y, z ∈A
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), ∀x, y, z ∈A
x ∧ y = x ∧ (x~ ∨ y), ∀x, y, z ∈A is called a Pre A*-algebra

Example (Venkateswara Rao, 2000): 3 = {0, 1, 2} with operations ∧, ∨, (-)~ defined below is a Pre A*-algebra.

Image for - Prime and Maximal Ideals of Pre A*-Algebra

Note: The elements 0, 1, 2 in the above example satisfy the following laws:

2~ = 2
(b) 1 ∧ x = x for all x ∈ 3
0 ∨ x = x for all x ∈ 3
2 ∧ x = 2 ∨ x = 2 for all x ∈ 3

Example: 2 = {0, 1} with operations ∧, ∨, (-)~ defined below is a Pre A*-algebra.

Image for - Prime and Maximal Ideals of Pre A*-Algebra

Note

(2, ∨, ∧, (-)~) is a Boolean algebra. So, every Boolean algebra is a Pre A* algebra
The identities 1.1(a) and 1.1(d) imply that the varieties of Pre A*-algebras satisfies all the dual statements of 1.1(b) to 1.1(g)

Note: Let, A be a Pre A*-algebra then A is Boolean algebra iff x∨(x∧y) = x, x∧ (x∨ y) = x (absorption laws holds).

Lemma 1: Venkateswara Rao and Srinivasa Rao (2009). Every Pre A*-algebra satisfies the following laws.

x∨ (x~∨ x) = x
(x∨x~)∧y = (x∧y) ∨ (x~∧y)
(x∨x~) ∧x = x
(x∨y)∧z = (x∧z)∨(x~∧y∧z)

Definition 3: (Venkateswara Rao and Srinivasa Rao, 2009): Let, A be a Pre A*-algebra. An element x ∈A is called central element of A if x∨x~ = 1 and the set {x ∈A/x∨x~ = 1} of all central elements of A is called the centre of A and it is denoted by B (A). Note that if A is a Pre A*-algebra with 1 then 1, 0∈B (A). If the centre of Pre A*-algebra coincides with {0, 1} then we say that A has trivial centre.

Theorem 1: Venkateswara Rao and Srinivasa Rao (2009). Let, A be a Pre A*-algebra with 1, then B (A) is a Boolean algebra with the induced operations ∧, ∨, (-)~.

Lemma 2: Venkateswara Rao and Srinivasa Rao (2009). Let, A be a Pre A*-algebra with 1:

If y ∈B(A) then x∧x~∧y = x∧x~, ∀ x∈A
x∧(x∨ y) = x∨(x∧y) = x if and only if x, yB∈(A)

IDEALS OF PRE A*-ALGEBRA

Definition 4: A nonempty subset U of a Pre A*-algebra A is said to be an ideal of A if the following hold:

a, b ∈ U ⇒a∨b∈U
a ∈U ⇒ x∧a∈U for each x∈A

Example: All the ideals of Pre A*-algebra A={0, 1, 2} are I1 ={2} I2={0, 2} and A itself.

Now we give some examples of Pre A*-algebras and collect all the ideals of these.

Example: Let, G={a1, a2, a3, a4, a5} where a1 =(1, 2), a2 = (0, 2), a3 = (2, 1), a4 = (2, 0), a5 = (2, 2). Then G is a Pre A*-algebra (a sub algebra of AxA) under the point wise operation. This algebra (G, ∧, ∨, (-)~) is a Pre A*-algebra without 1.All the ideals of G are”

I1={a5}

I2 = {a2, a5}
I3 = {a4, a5}
I4 = {a1, a2, a5}
I5 = {a2, a4, a5}
I6 = {a3, a4, a5}
I7 = {a1, a2, a4, a5}
I8 = {a2, a3, a4, a5}
I9 = G

Example: Let, H = {b1, b2, b3, b4, b5, b6, b7} where b1 =(1, 2), b2 = (0, 2), b3 = (2, 1), b4 = (2, 0), b5 = (2, 2), b6 =(1, 1), b7=(0, 0). Then H is a Pre A*-algebra (a sub algebra of AxA) under the point wise operation. This algebra (H, ∧, ∨, (-)~) is a Pre A*-algebra with 1 and 1= b6. All the ideals of H are:

I1 = {b5}
I2 = {b2, b5}
I3 = {b4, b5}
I4 = {b1, b2, b5}
I5 = {b3, b4, b5}
I6 = {b1, b2, b4, b5}
I7 = {b2, b3, b4, b5}
I8 = {b2,b4,b5, b7}
I9 = {b1,b2, b3, b4,b5}
I10 = {b1,b2, b4,b5, b7}
I11 = {b1, b2,b3, b4, b5, b7}
I12 = {b2,b3, b4, b5, b7}
I13 = H

Example: Let, F= AxA = {c1, c2, c3, c4, c5, c6, c7, c8, c9} where c1 =(1, 2), c2 = (0, 2), c3 = (2, 1), c4 = (2, 0), c5 = (2, 2), c6 =(1, 1), c7=(0, 0), c8=(1, 0), c9=(0, 1). Then F is a Pre A*-algebra under the point wise operation.

This algebra (F, ∧, ∨, (-)~) is a Pre A*-algebra with 1 and 1= c6. The ideals of AxA need not be of the form I1xI2 where, I1, I2 are ideals of A for such ideals of AxA are 9 in number (since, C has only 3 ideals namely {2}, {0, 2} and A) where, we are exhibiting 14 ideals in AxA.

All the ideals of F are:

I1 = {c5}
I2 = {c2, c5}
I3 = {c4, c5}
I4 = {c1, c2, c5}
I5 = {c3, c4, c5}
I6 = {c1, c2, c4, c5}
I7 = {c2, c3, c4, c5}
I8 = {c2, c4, c5, c7}
I9 = {c1, c2, c3, c4, c5}
I10 = {c1, c2, c4, c5, c7}
I11 = {c1, c2, c3, c4, c5, c7}
I12 = {c1, c2, c4, c5, c7, c8}
I13 = {c2, c3, c4, c5, c7, c9}
I14 = F

Lemma 3: Let, A be a Pre A*-algebra and p∈A. Then {x∧p /x∈A} is the smallest ideal containing p.

Proof: Let, U={x∧p /x∈A}. If s, t∈U then s= x∧p and t= y∧p for some x, y∈A s∨t=(x∧p) ∨ (y∧p)= (x∧p∧p) ∨ (y∧p∧p) = [(x∧p)∨ (y∧p)]∧p =(s∨t) ∧ p∈U. Therefore, s∨t∈U. Let, x∧p∈U and z∈A. Then, z∧(x∧p) = (z∧x)∧p∈U (since, z, x∧A∈z⇒x∈A). Therefore, U is an ideal of A. Also, U is an ideal containing p (since, p=p∧p∈U). Let, I be the ideal containing p. Then, xp∧I for any x∈A and hence U⊆I. Thus, U is the smallest ideal containing p.

Definition 5: Let, A be a Pre A*-algebra and p ∈A then {x∧p /x∈A} is called the Principal ideal generated by p and is denoted by < p >.

Now we give some examples of principal ideals of certain Pre A*-algebras.

Example: Principal ideals of the Pre A*-algebra A={0, 1, 2} are:

< 0 > = {0, 2}
< 1 > = {0, 1, 2}
< 2 > = {2}

Example: Principal ideals of the Pre A*-algebra G={a1, a2, a3, a4, a5} earlier are:

< a1 > = {a1,a2, a5}
< a2 > ={a2, a5}
< a3 > = {a3, a4, a5}
< a4 > = {a4, a5}
< a5 > = {a5}

Example: Principal ideals of the Pre A*-algebra H ={b1, b2, b3, b4, b5, b6, b7} earlier are:

< b1 > = {b1, b2, b5}
< b2 > = {b2, b5}
< b3 > = {b3, b4, b5}
< b4 > = {b4, b5}
< b5 > = {b5}
< b6 > = H
< b7 > = {b2, b4, b5, b7}

Example: Principal ideals of the Pre A*-algebra F= {c1, c2, c3, c4, c5, c6, c7, c8, c9} earlier are:

< c1 > = {c1, c2, c5}
< c2> = {c2, c5}
< c3 > = {c3, c4, c5}
< c4> ={c4, c5}
< c5 > = {c5}
< c6 > = F
< c7> ={c2, c4, c5, c7}
< c8 > = {c1, c2, c4, c5, c7, c8}
< c9 > = {c2, c3, c4, c5, c7, c9}

Lemma 4: Let, A be a Pre A*-algebra and p, y∈A then y∈< p >⇔y = y∧p.

Proof: Suppose, y∈< p > ⇒y= x∧p for some x∈A. Now y∧p= (x∧p)∧p = x∧p = y.

Conversely suppose that y = y∧p. Then, y∧p∈< p >.Therefore y∈< p >.

Lemma 5: Let, A be a Pre A*-algebra and p, q∈A then < p > = < q > if and only if p=q.

Proof: Suppose < p > = < q >. Clearly p∈< p > = < q > and q∈< q > =< p >:

⇒p= p∧q and q=q∧p (by lemma 5)
⇒p = q

On the other hand, p=q ⇒ p∧q=p and q∧p=q:

⇒p ∈< q > and q ∈< p >
⇒< p > ⊆ < q > and < q > ⊆ < p >
⇒< p > = < q >

Lemma 6: Let A be a Pre A*-algebra and p, q∈A. Then < p > ∩ < q > = < p∧q >.

Proof: Let, p, q∈A. Let, x∈< p∧q >⇒x=x∧ p∧q:

⇒x∧ p=x∧ p∧q∧ p=x∧ p∧q=x
⇒x∈< p >

Also, x∧q=x∧ p∧q∧q=x∧ p∧q=x:

⇒x∈< q >
⇒x∈< p > ∩ < q >

Therefore, <p∧q> ⊆ < p > ∩ < q >. Let, x∈< p >∩ < q >. Then, x∈< p > and x∈< q >:

⇒x=x∧ p and x=x∧q
⇒x=x∧q=(x∧ p)∧q= x∧ (p∧q)
⇒x∈< p∧q >

  Therefore, < p > ∩ < q >⊆< p∧q >. Thus, < p > ∩ < q > = < p∧q >.

Note: We do not have < p > ∪ < q > = < p∧q >, in general for example in the Pre A*-algebra G={a1, a2, a3, a4, a5} < a1 > ∪ < a3 >={a1,a2,a5} {a3, a4, a5}= G. But a1 ∨ a3= a5 and < a5 > ≠ G.

Definition 6: For any subset S of a Pre A*-algebra A and an ideal U, we define (S : U)={a∈A\ s∧a∈U for each s∈S}. If S={c} then we write it as (c : U)

Theorem 2: (S : U) is an ideal of A.

Proof: Let, a, b ∈ (S : U). Then s∧a, s∧b ∈ U for all s∈S. s∧(a∨b) = (s∧a) ∨ (s∧b) ∈ U (since, s∧a, s∧b∈U and U is ideal). Hence, a∨b∈ (S : U). Let, a ∈ (S : U), x∈A ⇒ a∧ s∈U for some s∈S. Then, (x∧a) ∧s=x∧(a ∧ s) ∈ U (since, a∧ s∈U, x∈A and U is ideal). Therefore, x∧a∈ (S : U). Thus, (S : U) is an Ideal of A.

Theorem 3: Let, S be a subset and U and W are the two ideals of Pre A*-algebra A then, (S : U∩W)= (S : U) ∩ (S : W).

Proof: Let, x∈ A. Then, x∈ (S : U∩W) ⇔ x ∧s ∈ U∩W, for all s∈S.

Image for - Prime and Maximal Ideals of Pre A*-Algebra

Therefore, (S : U∩W) = (S : U) ∩(S : W).

Theorem 4: Let, S and T are be any two subsets of Pre A*-algebra A and S ⊆ T then, (T : U) ⊆ (S :U).

Proof: Let, x∈(T : U) then x ∧t ∈ U for all t ∈ T and hence, x ∧s ∈ U for all s∈S (since S ⊆ T). So, that x ∈ (S : U). Therefore, (T : U) ⊆ (S : U).

Theorem 5: Let, S and T are be any two subsets of Pre A*-algebra A then, (S ∪ T: U) = (S : U)∩ (T: U).

Proof: We know that S and T ⊆ S ∩T and hence by lemma 6. (S ∪ T: U) ⊆ (S : U) ∩ (T : U). Now let, x∧ (S : U) ∩ (T : U). Then, x ∧ s ∈ U and x ∧ t ∈ U for all s ∈ S and t ∈T. Therefore, x ∧a∈U for all a ∈ S ∪T.

⇒x∈(S ∪T: U)

  Therefore, (S : U) ∩ (T : U) ⊆ (S ∪ T: U). Thus, (S ∪ T: U) = (S : U) ∩ (T : U).

Theorem 6: Let A be a Pre A*-algebra S⊆A for any ideals U and W of A < S >∩W⊆U ⇔W⊆ (S : U).

Proof: Suppose that W⊆ (S : U). Let, x∈< S >∩(S : U). Then:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

and x∈(S : U) where, si∈S, yi∈ A:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

and x∧si∩ U for each i. Now x = x ∧ x:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

Therefore, x ∈ U (since, x∧si∈ U). Therefore, < S >∩ (S : U) ∩ U and hence < S >∩W⊆ U (since, W⊆ (S : U)). On the other hand W is a ideal of A such that < S >∩W⊆U. Let, x∈W. Then, for any s ∈ S, x∧s∈< S >∩W (since W and < S > are ideals).

Yx∧s∈U (by supposition)

Therefore, x ∈ (S : U). Thus, W⊆ (S : U).

Lemma 7: Let, S be any subset of a Pre A*-algebra A. Then (< S > : U) = (S : U).

Proof: We know that S ⊆ < S > and therefore by theorem 4 (< S > : U) ⊆ (S : U). Let, x ∈(S:U).Then, x = x ∧s ∈ U for all s∈S. Let, t∈< S >. Then,

Image for - Prime and Maximal Ideals of Pre A*-Algebra

for some si ∈ S, yi ∈ A. Now,

Image for - Prime and Maximal Ideals of Pre A*-Algebra

Image for - Prime and Maximal Ideals of Pre A*-Algebra

  Then, x∧t∈ U. Thus, x∈(< S > : U). Therefore, (S : U) ⊆(< S > : U) and hence (< S > : U) = (S : U)

PRIME IDEALS OF PRE A*-ALGEBRA

Definition 7: Let, A be a Pre A*-algebra. A proper ideal of P of A is called a prime ideal if for any a, b∈A, a∧b∈P ⇒ either a ∈ P or b ∈ P.

Example: Consider the Pre A*-algebra A={0, 1, 2}. In this {0, 2} is the only prime ideal.

Example: The prime ideals of the Pre A*-algebra G={a1, a2, a3, a4, a5} (see 2.3 example) are I4={a1,a2,a5}, I6={a3, a4, a5},I7={a1, a2, a4, a5},I8={a2, a3, a4, a5} where as I1={a5}, I2={a2, a5}, I3={a4, a5}, I5={a2, a4, a5} are not prime ideals.

Example: Some of the prime ideals of the Pre A*-algebra H= {b1, b2, b3, b4, b5, b6, b7} (see 2.4 example) are, I10={b1,b2, b4,b5, b7}, I12={b2,b3, b4, b5, b7} where as I8={b2,b4,b5, b7}is an ideal but not a prime ideal.

Example: Some of the prime ideals of the Pre A*-algebra F= {c1, c2, c3, c4, c5, c6, c7, c8, c9} (see 2.5 example) are I12={c1, c2, c4, c5, c7, c8}, I13= {c2, c3, c4, c5, c7, c9} where as I8={c2, c4, c5, c7} is an ideal but not a prime ideal.

Theorem 7: The following are equivalent for any proper ideal P of Pre A*-algebra A

P is prime ideal
For any ideal I and J of A, I ∩ J ⊆ P⇒ I ⊆ P or J ⊆ P
For any ideal I and J of A, P= I ∩ J ⇒ P=I or P=J

Proof: (1) ⇒(2): Let, P be prime ideal and I and J are ideals of A such that I ∩J ⊆P. Assume that I é P. Then there exist a∈ I such that a ∉P. Let, b ∈ J. Then, a∧b ∈ I ∩ J ⊆ P (since, I ∈J ⊆P). Since, P is prime, a∈P or b∈P. But a∉P we have b∈P. Thus, J ⊆ P.

(2) ⇒ (3) is trivial
(3) ⇒ (1)

Suppose that I ∩ J=P ⇒ P=I or P=J for any ideals I and J of A. Let, a and b ∈A such that a∧b∈P. Then, by lemma 7 < a > ∩ < b > = < a ∧ b > ⊆ P and hence (< a >∨ P) ∩ (< b >∨P) = (< a > ∩ < b >) ∨ P = P so that < a > ∨ P =P or < b >∨P =P or equivalently < a >⊆P or < b >⊆P and therefore a ∈ P or b ∈ P. Thus, P is prime ideal.

Theorem 8: Let, I be an ideal of a Pre A*-algebra A and a∈A\I. Then there exist a prime ideal P containing I and not containing a.

Proof: Let, a∉ I. Let Image for - Prime and Maximal Ideals of Pre A*-Algebra = {J/J is an ideal of A, I ⊆ J and a ∉ J}. Clearly I∉Image for - Prime and Maximal Ideals of Pre A*-Algebra. Therefore, Image for - Prime and Maximal Ideals of Pre A*-Algebra is a nonempty and partially ordered set under set inclusion. By zorn’s lemma there exist a maximal member M in Image for - Prime and Maximal Ideals of Pre A*-Algebra. Then clearly M is a proper ideal of A and I⊆ M.

We shall prove that M is a prime ideal of A. Let x, y∈A such that x ∉M and y ∉M. Let, M is properly contained in M ∨< x > and M ∨< y > and hence by maximality of M, we have a∈ M ∨< x > ∩ M ∨< y > =M ∨(< x > ∧ < y >) = M ∨(< x∧y >).

Therefore, x∧y∉M (for if x∧y∈M then M ∨ (< x > ∧ < y >) = M which is a contradiction, since a ∉ M). Therefore, M is prime ideal containing I and not containing a.

Corollary: For any ideal I of a Pre A*-algebra A, I =∩{P / P is a prime ideal of A and I⊆P}.

Proof: If a∉ P then by theorem 8, there exist a prime ideal P containing I and not containing a and therefore, a:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

Thus, I =∩ {P / P is a prime ideal of A and I⊆P}

Theorem 9: Let, I be an ideal of a Pre A*-algebra A and S a nonempty subset of A which is closed under the operation ∧ and is disjoint with I. Then there exist a prime ideal P of A containing I and disjoint with S.

Proof: Let, I be an ideal of a Pre A*-algebra A, S⊆A and S∩I=φ. Let, Image for - Prime and Maximal Ideals of Pre A*-Algebra = {J/J is an ideal of A, I⊆J and J∩S=φ}.

Clearly I∈Image for - Prime and Maximal Ideals of Pre A*-Algebra. Therefore, Image for - Prime and Maximal Ideals of Pre A*-Algebra is a nonempty and partially ordered set under set inclusion. Image for - Prime and Maximal Ideals of Pre A*-Algebra is clearly closed under unions of chains. Therefore, by zorn’s lemma there exist a maximal member M in Image for - Prime and Maximal Ideals of Pre A*-Algebra. Then I ⊆ M and M ∩ S = φ.

We shall prove that M is prime ideal of A. Since, S is nonempty and M∩S = φ, M is proper ideal of A. Let, x, y∈A such that x∉M and y∉M.

Then M ∨< x > and M ∨< y > are ideals containing M properly. By the maximality of M, we get (M ∨ < x >) ∩ S ≠φ, (M ∨< y >) ∩ S ≠ φ.

Choose s ∈ (M ∨ < x >) ∩ S and t∈(M ∨< y >) ∩ S. Then, s ∧ t∈ S (since, S is closed under ∧ and s, t ∈ S) and s ∈ M ∨ < x > and t ∈ M ∨< y >.

Then, s ∧ t ∈ (M ∨< x >) ∩ (M∨< y >) = M ∨(< x > ∩ < y >)= M ∨(< x∧y >) and hence, s∧t∈ M ∨ (< x ∧ y >) ∩ S.

Then it follows that x∧y∉M (for, if x∧y∈M then M ∨ (< x∧y >) = M and hence s∧t∈M∩S, which is contradiction to the fact that M∩ S = φ)

Thus, M is the prime ideal of A, I⊆M and M∩S = φ.

Theorem 10: For any ideals I and J of Pre A*-algebra A, (I: J) = ∩ {P / P is a prime ideal of A, J ⊆P and I éP}.

Proof: We have I∩(I: J) ⊆ J. If P is prime ideal containing J then I ∩ (I: J) ⊆ J ⊆ P and if in addition I é P then (I: J) ⊆ P. Therefore, (I: J) ⊆ ∩ {P/P is a prime ideal of A, J ⊆ P and I é P}. On the other hand, suppose x ∉ (I: J).

Then there exist a ∈ I such that x ∧ a ∉ J. Therefore, there exists prime ideal P such that J ⊆ P and x ∧ a ∉ P. Then, x ∉ P and a ∉ P (since, P is prime) and hence, I é P (since, a ∈ I and a ∉ P). Therefore, x ∉1 {P / P is a prime ideal of A, J⊆P and IéP} and hence, ∩{P / P is a prime ideal of A, J⊆P and I é P ⊆ (I: J). Thus, (I: J) = ∩ {P / P is a prime ideal of A, J ⊆ P and I éP}.

MAXIMAL IDEALS OF PRE A*-ALGEBRA

Definition 8: Let A be a Pre A*-algebra. A proper ideal of M of A is called a Maximal ideal of A if M is maximal among the proper ideals of A.

Lemma 8: Let A be a Pre A*-algebra. Every maximal ideal in A is a prime ideal.

Proof: Let M is a maximal ideal of A. Then clearly M is a proper ideal of A. Suppose a, b∈A such that a ∉ M and b ∉M.

Then, M Image for - Prime and Maximal Ideals of Pre A*-Algebra< M ∪ {a} > = A; M Image for - Prime and Maximal Ideals of Pre A*-Algebra< M ∪ {b} > =A (since M is maximal). Then:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

since, b ∈A we have b ∈ < M ∪{a}.

That is:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

for some yi∈A, xi ∈ M ∪{a}. Now b=b ∧ b:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

If xi ∈ M then clearly b ∧ yi ∧ xi ∈ M. Since, b ∉ M and

Image for - Prime and Maximal Ideals of Pre A*-Algebra

it follows that b ∧ yi∧a∉M for some yi (since, xi∈ M ∪ {a} ⇒ xi =M or xi =a). Therefore, b∧a∉M⇒a∧b∉M (since M is ideal).

Thus, M is prime ideal of A. The validity of the converse of the theorem 10 is not known. That is, are there prime ideals which are not maximal. In certain special cases, we know that every prime ideal is maximal, for example Boolean algebras. In the following we discuss another class of Pre A*-algebra when every prime ideal is maximal.

Theorem 11: Let, A be a Pre A*-algebra with 1.Suppose that for any x ∈A, there exists a smallest x0∈B(A) such that x∧ x0 = x and x0~∧x = 1. Then, every prime ideal of A is a maximal ideal.

Proof: Let, P is a prime ideal of A and Q be any ideal such that P Image for - Prime and Maximal Ideals of Pre A*-AlgebraQ. Since, P Image for - Prime and Maximal Ideals of Pre A*-AlgebraQ there exits x∈Q such that x ∉P.

Then there exists a smallest x0 ∈ B(A) such that x∧ x0 = x and x0~∨x = 1. Therefore, x0∉P (for if x0 ∈ P then x ∈ P, a contradiction).

Since, P is prime x0~∈ P⊆Q. Therefore both x and x0~ belongs to Q and hence, 1= x0~∨x∈Q (since, Q is ideal).

Then a=1∧ a∈Q for all a∈A (since, 1∈Q, a∈ A and Q is ideal, 1∧ a ∈Q). Therefore, Q=A. Thus P is maximal ideal of A.

Definition 9: A prime ideal P of a Pre A*-algebra A is said to be a minimal prime ideal of A if there is no prime ideal Q of A such that Q ⊂P.

Lemma 9: Let, A be a Pre A*-algebra. Every prime ideal contains a minimal prime ideal.

Proof: Let, P be a minimal prime ideal of A. Let, Image for - Prime and Maximal Ideals of Pre A*-Algebra = {Q / Q is a prime ideal of A, Q⊆P}. Then,Image for - Prime and Maximal Ideals of Pre A*-Algebra is a nonempty (P∈Image for - Prime and Maximal Ideals of Pre A*-Algebra) and T is a partially ordered set with respect to inclusion. If {Qα}αεΔ is a chain in Image for - Prime and Maximal Ideals of Pre A*-Algebra then:

Image for - Prime and Maximal Ideals of Pre A*-Algebra

in Image for - Prime and Maximal Ideals of Pre A*-Algebra.

Therefore every chain in T has a lower bound in Image for - Prime and Maximal Ideals of Pre A*-Algebra. Thus, by zorn’s lemma Image for - Prime and Maximal Ideals of Pre A*-Algebra has a minimal member say M. Then M is a prime ideal of A and M ⊆ P. Suppose there is a prime ideal R of A such that R ⊆ M. Then, R ⊆ M ⊆ P. Therefore, R ⊆ Image for - Prime and Maximal Ideals of Pre A*-Algebra. By the minimality of M in Image for - Prime and Maximal Ideals of Pre A*-Algebra we get that R=M. Hence, M is the minimal prime ideal contained in P.

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