| Title:
|
A note on generalizations of semisimple modules (English) |
| Author:
|
Kaynar, Engin |
| Author:
|
Türkmen, Burcu N. |
| Author:
|
Türkmen, Ergül |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
3 |
| Year:
|
2019 |
| Pages:
|
305-312 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A left module $M$ over an arbitrary ring is called an $\mathcal{RD}$-module (or an $\mathcal{RS}$-module) if every submodule $N$ of $M$ with ${\rm Rad}(M)\subseteq N$ is a direct summand of (a supplement in, respectively) $M$. In this paper, we investigate the various properties of $\mathcal{RD}$-modules and $\mathcal{RS}$-modules. We prove that $M$ is an $\mathcal{RD}$-module if and only if $M={\rm Rad}(M)\oplus X$, where $X$ is semisimple. We show that a finitely generated $\mathcal{RS}$-module is semisimple. This gives us the characterization of semisimple rings in terms of $\mathcal{RS}$-modules. We completely determine the structure of these modules over Dedekind domains. (English) |
| Keyword:
|
radical |
| Keyword:
|
supplement |
| MSC:
|
16D10 |
| MSC:
|
16D99 |
| idZBL:
|
Zbl 07144896 |
| idMR:
|
MR4034434 |
| DOI:
|
10.14712/1213-7243.2019.011 |
| . |
| Date available:
|
2019-10-29T12:52:42Z |
| Last updated:
|
2021-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147860 |
| . |
| Reference:
|
[1] Alizade R., Bilhan G., Smith P. F.: Modules whose maximal submodules have supplements.Comm. Algebra 29 (2001), no. 6, 2389–2405. MR 1845118, 10.1081/AGB-100002396 |
| Reference:
|
[2] Büyükaşik E., Pusat-Yilmaz D.: Modules whose maximal submodules are supplements.Hacet. J. Math. Stat. 39 (2010), no. 4, 477–487. MR 2796587 |
| Reference:
|
[3] Büyükaşik E., Türkmen E.: Strongly radical supplemented modules.Ukrainian Math. J. 63 (2012), no. 8, 1306–1313. MR 3109654 |
| Reference:
|
[4] Lomp C.: On semilocal modules and rings.Comm. Algebra 27 (1999), no. 4, 1921–1935. MR 1679679, 10.1080/00927879908826539 |
| Reference:
|
[5] Nebiyev C., Pancar A.: On supplement submodules.Ukrainian Math. J. 65 (2013), no. 7, 1071–1078. MR 3145891, 10.1007/s11253-013-0842-2 |
| Reference:
|
[6] Türkmen B. N., Pancar A.: Generalizations of $\oplus$-supplemented modules.Ukrainian Math. J. 65 (2013), no. 4, 612–622. MR 3125012, 10.1007/s11253-013-0799-1 |
| Reference:
|
[7] Türkmen B. N., Türkmen E.: On a generalization of weakly supplemented modules.An. Ştiin. Univ. Al. I. Cuza Din Iaşi. Mat. (N.S.) 63 (2017), no. 2, 441–448. MR 3718613 |
| Reference:
|
[8] Wisbauer R.: Foundations of Module and Ring Theory.A handbook for study and research, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, 1991. Zbl 0746.16001, MR 1144522 |
| . |
