Some conditions for finiteness and commutativity of rings

Abstract

ABSTRACT. We In what follows, R is an associative ring with center C. The set of nilpotent elements is denoted by N; and for a subset S of R, the subring generated by S is denoted by <S>. The term zero divisor will mean a one-sided zero divisor (i.e. not necessarily a two-sided zero divisor), and 0 will be considered a zero divisor. For x e R, the symbols Ar(X A (x), and A(x) denote respectively the right, left, and two-sided annihilators of x. Finally, the symbols Z, Z n, and C(p) denote respectively the ring of integers, the ring of integers mod. n, and the Prefer p-group.We shall frequently have use for direct-sun deccmpositions of R, both as a ring and as an additive group. To make the distinction clear, we use the symbol to denote a ring-theoretic direct sun and the symbol to denote an additive-group direct sum. 2. A FINITENESS RESULT.We begin by discussing rings in which certain subrings of zero divisors are assuned to be finite.

Which Fields Have No Maximal Subrings?

“…In [5], it is proved that if R is a finite maximal subring of a ring T, then T must be finite too. This is a trivial consequence of the fact that whenever R is a maximal subring of T, then R is Artinian if and only if T is Artinian and integral over R, see [3, PROOF.…”

“…We denote the set of maximal subrings (maximal ideals) of a ring by RgMax(R) (Max(R)). Maximal subrings are also studied in [17], [5], [6], [14], [13], [16], [2], [3] and [4]. In [2], maximal subrings of commutative rings are investigated and some useful criterions for the existence of maximal subrings are given.…”

On finiteness of rings with finite maximal subrings

“…It was proved by Bell and Guerriero [4] that a ccmatative ring with a finite maximal subring is finite, and it is our purpose to extend this result to pI-rings. The full force of the PI assumption is used only once in the proof of our theorem; the proofs of the lns use only the fact that the class of pI-rings is closed under taking subrings and hcmcmorphic images.…”

“…The subring generated by T is denoted by <T>. by recalling a crucial result frcm [4], which is obtained by applying an interesting result of Lewin [5 ].…”

“…As in [4] In proving that I is a simple ring, we first sh that 12 0. We prove that 12 0 implies R is finite.…”