On semi-simple radical classes

Abstract

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) .

Essentially closed radical classes

“…It is well-known that a homomorphically closed semisimple class is a radical class (see Wiegandt [6] Theorem 32.1). Moreover if C is not the class of all rings, then every C-ring is a subdirect sum of finite fields (see Stewart [4] Let us note that all homomorphically closed semisimple classes have been explicitly determined (see Gardner and Stewart [2]). There are countably many such classes and each of them is determined by a strictly hereditary finite set of finite fields (see Stewart [4] …”

Axiomatic radical and semisimple classes of rings

“…(ii) The radical-semisimple classes V(P, N) characterized in [6]. These radical classes are subvarieties of rings.…”

Semi-simple radical classes of algebras and attainability of identities

“…Shevrin and Martynov [25] showed that the nontrivial varieties of associative rings with attainable identities are precisely the varieties generated by finite sets of finite fields, while Martynov [19] proved that these are precisely the nontrivial idempotent varieties. For a purely radical-theoretic account of semi-simple radical classes in this context see [31] and [9]. In particular, it is shown in [9] that the classes T n = {A I a n = aVa e A} , n = 2, 3, 4, are semi-simple radical classes, though there are others.…”

“…For a purely radical-theoretic account of semi-simple radical classes in this context see [31] and [9]. In particular, it is shown in [9] that the classes T n = {A I a n = aVa e A} , n = 2, 3, 4, are semi-simple radical classes, though there are others. It has been widely accepted ( [17], [30], [35], [36]) that the T n are the only (nontrivial) semi-simple radical classes.…”