On Subdirect Decomposition and Varieties of Some Rings with Involution. Ii

As it is clearly suggested by the title, this note is a continuation of [1]. In the latter paper, the authors start from the famous theorem of N. Jacobson which asserts that every ring satisfying the identity x = x for some n ≥ 1 must be commutative (though Jacobson’s result is more general: the existence of a positive integer n(a) for each a ∈ R such that a = a suffices to conclude that the ring R is commutative). One way (which is, for obvious reasons, quite popular among universal algebraists) to see this is to determine, for a fixed n, the subdirectly irreducible rings with the identity x = x, e.g. as in [4, pp.175–178]. It turns out that these subdirectly irreducibles are precisely the finite fields Fpk such that (pk−1) | n. Hence, every ring satisfying an identity of the form x = x is a subdirect product of finite fields, and thus commutative. Motivated by this approach, in [1] all subdirectly irreducible involution rings satisfying x = x were determined. Recall that an involution ring is a structure (R, ∗) such that R is a ring, and the unary operation ∗ is an involutorial antiautomorphism of R, i.e. we have (x + y)∗ = x∗ + y∗, (xy)∗ = y∗x∗ and (x∗)∗ = x (we refer e.g. to [2, 3, 6, 7] for an overview of involution rings). The result is as follows (the notation is slightly changed, but is still standard).