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On the distributivity of the lattice of radical submodules | ||
| Journal of Mahani Mathematical Research | ||
| دوره 13، شماره 1 - شماره پیاپی 26، بهمن 2023، صفحه 347-355 اصل مقاله (478.09 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22103/jmmr.2023.21494.1439 | ||
| نویسندگان | ||
| Hossein Fazaeli Moghimi* ؛ Morteza Noferesti | ||
| Department of Mathematics, University of Birjand, Birjand, Iran | ||
| چکیده | ||
| Let $R$ be a commutative ring with identity and $\mathcal{R}(_{R}M)$ denotes the bounded lattice of radical submodules of an $R$-module $M$. We say that $M$ is a radical distributive module, if $\mathcal{R}(_{R}M)$ is a distributive lattice. It is shown that the class of radical distributive modules contains the classes of multiplication modules and finitely generated distributive modules properly. It is shown that if $M$ is a semisimple $R$-module and for any radical submodule $N$ of $M$ with direct sum complement $\tilde{N}$, the complementary operation on $\mathcal{R}(_{R}M)$ is defined by $N':=\tilde{N}+rad(0)$, then $\mathcal{R}(_{R}M)$ with this unary operation forms a Boolean algebra. In particular, if $M$ is a multiplication module over a semisimple ring $R$, then $\mathcal{R}(_{R}M)$ is a Boolean algebra, which is also a homomorphic image of $\mathcal{R}(_{R}R)$. | ||
| کلیدواژه‌ها | ||
| Radical distributive module؛ distributive module؛ multiplication module؛ semisimple ring؛ Boolean algebra homomorphism | ||
| مراجع | ||
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