Some notes on nil-semicommutative rings

On Some Generalizations of Reversible and Semicommutative Rings

The concept of strongly central reversible rings has been introduced in this paper. It has been shown that the class of strongly central reversible rings properly contains the class of strongly reversible rings and is properly contained in the class of central reversible rings. Various properties of the above-mentioned rings have been investigated. The concept of strongly central semicommutativ...

Some notes on CN rings

The main results: A ring R is CN if and only if for any x ∈ N(R) and y ∈ R, ((1+x)y)n+k = (1+x)n+kyn+k, where n is a fixed positive integer and k = 0, 1, 2; (2) Let R be a CN ring and n ≥ 1. If for any x, y ∈ R\N(R), (xy)n+k = xn+kyn+k, where k = 0, 1, 2, then R is commutative; (3) Let R be a ring and n ≥ 1. If for any x ∈ R\N(R) and y ∈ R, (xy)k = xkyk, k = n, n + 1, n + 2, then R is commutati...

Some Notes on Semiabelian Rings

Strongly nil-clean corner rings

We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings‎, ‎then $R/J(R)$ is nil-clean‎. ‎In particular‎, ‎under certain additional circumstances‎, ‎$R$ is also nil-clean‎. ‎These results somewhat improves on achievements due to Diesl in J‎. ‎Algebra (2013) and to Koc{s}an-Wang-Zhou in J‎. ‎Pure Appl‎. ‎Algebra (2016)‎. ‎...

On primitive ideals in polynomial rings over nil rings

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I [x] for some ideals I of R. All considered rings are associative but not necessarily have identities. Köthe’s conjecture states that a ring without nil ideals has no one-sided nil ideals. It is equivalent [4] to the assertion that polynomial rings over nil rings are Jaco...