*-ZERO DIVISORS AND *-PRIME IDEALS

Abstract

Throughout this note we introduce the concept of *-zero divisors in rings with involution and its correlation with the concept of zero divisors in rings without involution. Moreover, some related definitions; such as *-completely prime ideals and rings and *-cancellation laws are introduced. Nevertheless, we characterize *-prime and *-completely prime ideals using *-zero divisors. By a ring we mean an associative ring. A ring A is said to be an involution ring if on A there is defined a unary operation (called involution) ∗ subject to the identities a∗∗= a,(a+b)∗= a∗+b∗ and (ab)∗= b∗a∗, for all a,b∈A . In other words, the involution is an anti-isomorphism of order 2 on A . For a commutativering A , it isevident that the identitymapping of A onto A is an involution on A (see [1]-[4]) . Considering the category of involution rings, all morphisms (and also embeddings) must preserve involution. So we are looking here for a paricular concept for zero divisors that works in the category of involution rings. Iftheideal I ofA isclosedunder involution;thatisI(∗)={a∗∈A | a∈ I}⊆ I, then it is called a *-ideal of A and will be denoted by I ∗ A . We start by defining *-zero divisors for an involution ring A . Definition 1 A nonzero element a ∈ A is said to be a *-zero divisor if there exists a nonzero element b∈A such that ab = 0 anda∗b =0 . Remark 2 If we start by defining left *-zero divisor as in definition 1, we get b∗a∗= 0 and b∗a = 0 which mean thata is a right *-zero divisor, too. By reversing the roles, a right *-zero divisor is also a left *-zero divisor. Thus we have only the concept of *-zero divisor, as one expects in the category of involution rings.