Finite separability in varieties of associative rings
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A subset M of a universal algebra A is called finitely separated in A if, for any element x∈A/M, there exists an homomorphism ϕ of A into a finite algebra, for which ϕ(x)∋ϕ(M). A ring is said to be S-separable (R-separable) if its subrings (resp., right ideals) are all finitely separated in it. We give equational (in the language of identities) and indicator (in the language of “prohibited” rings) characterizations of varieties consisting of S-separable (R-separable) rings. Moreover, varieties are described in which, not all, but finitely generated rings only share the properties mentioned.
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