Commutativity of One Sided S- Unital Rings

Abstract

ABSTRACT. The following theorem is proved: Let r r(y) > 1, s, and be non-negative integers. If R is a left s-unital ring satisfies the polynomial identity [xy x'y"x t, x] 0 for every x, y E R, then R is commutative. The commutativity of a right s-unital ring satisfying the polynomial identity [x/-yrxt, X] 0 for all x, y E R, is also proved.

On commutativity of rings with constraints involving a nil subset

“…In fact, severa! On Commutativity of Rings with Constraints commutativity theorems can be obtained as a corollaries to our results (see [1], [2] and [5]). (e) R satisfies the condition (p2) and there exists a nil subset B of R for which R satisfies ( **-B).…”

“…Example 4 shows that the above conjec:ture is not true beca use the c:entrality of idempotent in S 1 ( resp 8 2 ) are not followed by (pi) (resp. (p 2 )) together with (**-B).…”

“…rings satisfying ( *-B) under some restrictions on B. Many authors have studied the comnmtativity of rings satisfying tlw property ( *-B), bu t always under sorne restriction on B. Severa] special cases of (PI) and (P2) are known to imply commutativity of rings (see [1], [:2] and [9]). For instance, if the integral índices in the underlying conditions are globaL The major purpose of this paper is to investigate the commutativity of R when the integral indices are local (i.e.…”

Commutativity of rings with polynomial constraints

“…(P 1 ) For each x ∈ R, there exist polynomials f (λ) in λ 2 [λ] and g(λ), h(λ) in (P 3 ) Let p, q and r be fixed non-negative integers. For each x, y ∈ R there exists a polynomial f (λ) ∈ λ 2 [λ] such that…”

“…Now, we consider the following ring properties: (P) For each x in R, there exist polynomials f (λ) ∈ λ 2 [λ] and g(λ), h(λ) ∈ [λ]…”