On Zero Subrings and Periodic Subrings
نویسنده
چکیده
We give new proofs of two theorems on rings in which every zero subring is finite; and we apply these theorems to obtain a necessary and sufficient condition for an infinite ring with periodic additive group to have an infinite periodic subring. 2000 Mathematics Subject Classification. 16N40, 16N60, 16P99. Let R be a ring and N its set of nilpotent elements; and call R reduced if N = {0}. Following [4], call R an FZS -ring if every zero subring—that is, every subring with trivial multiplication—is finite. It was proved in [1] that every nil FZS -ring is finite—a result which in more transparent form is as follows. Theorem 1. Every infinite nil ring contains an infinite zero subring. Later, in [4], it was shown that every ring with N infinite contains an infinite zero subring. The proof relies on Theorem 1 together with the following result. Theorem 2 (see [4]). If R is any semiprime FZS-ring, then R = B⊕C , where B is reduced and C is a direct sum of finitely many total matrix rings over finite fields. Theorems 1 and 2 have had several applications in the study of commutativity and finiteness. Since the proofs in [1, 4] are rather complicated, it is desirable to have new and simpler proofs; and in our first major section, we present such proofs. In our final section, we apply Theorems 1 and 2 in proving a new theorem on existence of infinite periodic subrings. 1. Preliminaries. Let Z and Z+ denote, respectively the ring of integers and the set of positive integers. For the ring R, denote by the symbols T and P(R), respectively the ideal of torsion elements and the prime radical; and for each n∈ Z+, define Rn to be {x ∈ R | xn = 0}. For Y an element or subset of R, let 〈Y 〉 be the subring generated by Y ; let Al(Y), Ar(Y), and A(Y) be the left, right, and two-sided annihilators of Y ; and let CR(Y) be the centralizer of Y . For x,y ∈ R, let [x,y] be the commutator xy−yx. The subring S of R is said to be of finite index in R if (S,+) is of finite index in (R,+). An element x ∈ R is called periodic if there exist distinct positive integers m, n such that xm = xn; and the ring R is called periodic if each of its elements is periodic. We will use without explicit mention two well-known facts: (i) the intersection of finitely many subrings of finite index in R is a subring of finite index in R;
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