Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
radical; supplement
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.
References:
[1] Alizade R., Bilhan G., Smith P. F.: Modules whose maximal submodules have supplements. Comm. Algebra 29 (2001), no. 6, 2389–2405. DOI 10.1081/AGB-100002396 | MR 1845118
[2] Büyükaşik E., Pusat-Yilmaz D.: Modules whose maximal submodules are supplements. Hacet. J. Math. Stat. 39 (2010), no. 4, 477–487. MR 2796587
[3] Büyükaşik E., Türkmen E.: Strongly radical supplemented modules. Ukrainian Math. J. 63 (2012), no. 8, 1306–1313. MR 3109654
[4] Lomp C.: On semilocal modules and rings. Comm. Algebra 27 (1999), no. 4, 1921–1935. DOI 10.1080/00927879908826539 | MR 1679679
[5] Nebiyev C., Pancar A.: On supplement submodules. Ukrainian Math. J. 65 (2013), no. 7, 1071–1078. DOI 10.1007/s11253-013-0842-2 | MR 3145891
[6] Türkmen B. N., Pancar A.: Generalizations of $\oplus$-supplemented modules. Ukrainian Math. J. 65 (2013), no. 4, 612–622. DOI 10.1007/s11253-013-0799-1 | MR 3125012
[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
[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. MR 1144522 | Zbl 0746.16001
Partner of
EuDML logo