Nilpotent elements and Armendariz rings

Nilpotent Elements in Skew Polynomial Rings

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...

Properties of Armendariz Rings and Weak Armendariz Rings

We consider some properties of Armendariz and rigid rings. We prove that the direct product of rigid (weak rigid), weak Armendariz rings is a rigid (weak rigid), weak Armendariz ring. On the assumption that the factor ring R/I is weak Armendariz, where I is nilpotent ideal, we prove that R is a weak Armendariz ring. We also prove that every ring isomorphism preserves weak skew Armendariz struct...

On Classical Quotient Rings of Skew Armendariz Rings

Let R be a ring, α an automorphism, and δ an α-derivation of R. If the classical quotient ring Q of R exists, then R is weak α-skew Armendariz if and only if Q is weak α-skew Armendariz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly ci...

Extensions of Nilpotent P.P. Rings

On quasi-Armendariz skew monoid rings

Let $R$ be a unitary ring with an endomorphism $σ$ and $F∪{0}$ be the free monoid generated by $U={u_1,…,u_t}$ with $0$ added, and $M$ be a factor of $F$ setting certain monomial in $U$ to $0$, enough so that, for some natural number $n$, $M^n=0$. In this paper, we give a sufficient condition for a ring $R$ such that the skew monoid ring $R*M$ is quasi-Armendariz (By Hirano a ring $R$ is called...