It has been incorrectly asserted that each non-trivial semi-simple radical class of associative rings is a variety defined by an equation of the form x = x . In this paper we give, for each non-trivial semi-simple radical class of associative rings, a set of equations which does define that class as a variety.W e shall discuss conditions on a class C of associative rings which are equivalent to C being a semi-simple radical class. See [7] for a discussion of the more general case in which C is a class of algebras which are not necessarily associative rings. Our theorem corrects an assertion which appears to have been first made by Snider [6] and which has been widely accepted by other authors.If a is an element of a ring R , [a] denotes the subring of R which is generated by a , and the class of rings R such that [a] = [a] for each a € R is denoted by B .Let P be a finite non-empty set of prime numbers and, for each p € P , N{p) a finite non-empty set of positive integers. The equations where p = Tl{q £ P ; q ± p} , define a variety which we shall denote by
V(P, N) .
A ring R is prime essential if R is semiprime and for each prime ideal P of R, P n / # 0 whenever / is a nonzero two-sided ideal of R. Examples of prime essential rings include rings of continuous functions and infinite products modulo infinite sums. We show that the class of prime essential rings is closed under many familiar operations; in particular, we consider polynomial rings, matix rings, fixed rings and skew group rings. Also, we explore the relationship between prime essential rings and special radical classes, and we demonstrate how prime essential rings can be used to construct radical classes which are not special.
Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.
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