On three problems concerning nil-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...

Commutative Nil Clean Group Rings

In [5] and [6], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short article we characterize nil clean commutative group rings.

On three problems concerning the Smarandache LCM sequence

In this paper three problems posed in [1J and concerning the Smarandache LeM sequence have been analysed. Introduction In [1] the Smarandache LCM . sequence is defined as the least common mUltiple (LCM) of (1,2,3, .,. ,n) : 1,2,6, 12,60,60,420,840,2520,2520,27720,27720,360360, 360360, 360360, 720720 ...... . In the same paper the following three problems are reported: 1. If a(n) is the n-th ter...

Some problems and results related to nil and graded-nilpotent rings