On commutativity theorems for rings

Abstract

Let R be an associative rlng with unity. It is proved that if R satisfies Ohe polynomial identity Ix n y ymxn, x] 0 (m I, n > I), then R is commutative. Two or more related results are also obtained.

Commutativity of One Sided S- Unital Rings

“…THEOREM HK ( [1,Theorem] r(z,u"') > 1, s2 s(x,u') >_ 0, and t =.t(x,u ') >_ O, then (3.3) It is well-known that R is isomorphic to a subdirect sum of subdirectly irreducible rings Ri (i 6-I, the index set), each of which as a homomorphic image of R satisfies the property placed on R. Thus R itself cart be assumed to be subdirectly irreducible ring. Let S be the intersection of all its non-zero ideals of R. So S = (0).…”