Central Armendariz Rings

Nazim Agayev ; Gonca Güngöroglu ; A. Harmanci ; S. Halicioglu

Bulletin of the Malaysian Mathematical Society, Tome 34 (2011) no. 1

Résumé

We introduce the notion of central Armendariz rings which are a generalization of Armendariz rings and investigate their properties. We show that the class of central Armendariz rings lies strictly between classes of Armendariz rings and abelian rings. For a ring $R$, we prove that $R$ is central Armendariz if and only if the polynomial ring $R[x]$ is central Armendariz if and only if the Laurent polynomial ring $R[x, x^{-1}]$ is central Armendariz. Moreover, it is proven that if $R$ is reduced, then $R[x]/(x^{n})$ is central Armendariz, the converse holds if $R$ is semiprime, where $(x^n)$ is the ideal generated by $x^n$ and $n\geq 2$. Among others we also show that $R$ is a reduced ring if and only if the matrix ring $T_{n}^{n-2}(R)$ is central Armendariz, for a natural number $n \geq 3$ and $k=[n/2]$.

Détail

Classification : 16U80.