Semicommutativity of the rings relative to prime radical

Kose, Handan, and Ungor, Burcu. "Semicommutativity of the rings relative to prime radical." Commentationes Mathematicae Universitatis Carolinae 56.4 (2015): 401-415. <http://eudml.org/doc/276260>.

@article{Kose2015,
abstract = {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.},
author = {Kose, Handan, Ungor, Burcu},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semicommutative ring; $P$-semicommutative ring; prime radical of a ring},
language = {eng},
number = {4},
pages = {401-415},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Semicommutativity of the rings relative to prime radical},
url = {http://eudml.org/doc/276260},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Kose, Handan
AU - Ungor, Burcu
TI - Semicommutativity of the rings relative to prime radical
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 4
SP - 401
EP - 415
AB - 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.
LA - eng
KW - semicommutative ring; $P$-semicommutative ring; prime radical of a ring
UR - http://eudml.org/doc/276260
ER -