Some notes on nil-semicommutative rings

Abstract

A ring R is defined to be nil-semicommutative if ab ∈ N (R) implies arb ∈ N (R) for a, b, r ∈ R , where N (R) stands for the set of nilpotents of R . Nil-semicommutative rings are generalization of N I rings. It is proved that (1) R is strongly regular if and only if R is von Neumann regular and nil-semicommutative; (2) Exchange nil-semicommutative rings are clean and have stable range 1; (3) If R is a nil-semicommutative right M C2 ring whose simple singular right modules are Y J− injective, then R is a reduced weakly regular ring; (4) Let R be a nil-semicommutative π− regular ring. Then R is an (S, 2) -ring if and only if Z/2Z is not a homomorphic image of R .

Some notes on $GQN$ rings

“…Recall that a ring R is nil -semicommutative [7] if ab ∈ N (R) implies arb ∈ N (R) for all a, b, r ∈ N (R).…”

“…Hence, there exists a positive integer m such that (gt) m = 0. Recall that a ring R is directly finite if ab = 1 implies ba = 1 for all a, b ∈ R , and R is said to be n regular [12] (2) and (3) are direct corollaries of (1 Recall that a ring R is nil -semicommutative [7] if ab ∈ N (R) implies arb ∈ N (R) for all a, b, r ∈ N (R). …”

Semicommutativity of the rings relative to prime radical

“…Every exchange ring with all idempotents central is clean (see [15] for detail). In [17], it is proved that every nil-semicommutative-II exchange ring is clean. We now extend these results to P -semicommutative rings.…”

“…There are nil-semicommutative-I rings that are not semicommutative. Another type of nil-semicommutative rings is defined in [17] and [6]. Again to get rid of confusion, we call this nil-semicommutative ring nil-semicommutative-II.…”

NEC rings

“…[21, Proposition 5.3] An exchange ring R has stable range one if and only if for each regular element a of R, there exists u ∈ U(R) such that a − aua ∈ J(R).…”