| Title:
|
Semicommutativity of the rings relative to prime radical (English) |
| Author:
|
Kose, Handan |
| Author:
|
Ungor, Burcu |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
401-415 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
semicommutative ring |
| Keyword:
|
$P$-semicommutative ring |
| Keyword:
|
prime radical of a ring |
| MSC:
|
16S50 |
| MSC:
|
16U99 |
| idZBL:
|
Zbl 06537716 |
| idMR:
|
MR3434221 |
| DOI:
|
10.14712/1213-7243.2015.140 |
| . |
| Date available:
|
2015-12-17T11:43:31Z |
| Last updated:
|
2018-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144750 |
| . |
| Reference:
|
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