Semicommutativity of the rings relative to prime radical

Kose, Handan ; Ungor, Burcu

Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 4, pp. 401-415.

Résumé

In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$-semicommutative. We prove that a ring $R$ is $P$-semicommutative if and only if $R[x]$ is $P$-semicommutative if and only if $R[x, x^{-1}]$ is $P$-semicommutative. Also, if $R[[x]]$ is $P$-semicommutative, then $R$ is $P$-semicommutative. The converse holds provided that $P(R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$, $R$ is $P$-semicommutative if and only if $T_n(R)$ is $P$-semicommutative. For a ring $R$ of bounded index $2$ and a central nilpotent element $s$, $R$ is $P$-semicommutative if and only if $K_s(R)$ is $P$-semicommutative. If $T$ is the ring of a Morita context $(A,B,M,N,\psi,\varphi)$ with zero pairings, then $T$ is $P$-semicommutative if and only if $A$ and $B$ are $P$-semicommutative. Many classes of such rings are constructed as well. We also show that the notions of clean rings and exchange rings coincide for $P$-semicommutative rings.

Détail

DOI : 10.14712/1213-7243.2015.140

Classification : 16S50, 16U99
Keywords: semicommutative ring; $P$-semicommutative ring; prime radical of a ring