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arXiv · 2208.07940

Cyclic-Uniform Uniserial Modules and Rings

R. Nikandish·M. J. Nikmehr·A. Yassine
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Abstract

An $R$-module $M$ is called virtually uniserial if for every finitely generated submodule $0 \neq K \subseteq M$, $K/$Rad$(K)$ is virtually simple. In this paper, we generalize virtually uniserial modules by dropping the virtually simple condition and replacing it by the cyclic uniform condition. An $R$-module $M$ is called cyclic-uniform uniserial if $K/$Rad$(K)$ is cyclic and uniform, for every finitely generated submodule $0 \neq K \subseteq M$. Also, $M$ is said to be cyclic-uniform serial if it is a direct sum of cyclic-uniform uniserial modules. Several properties of cyclic-uniform (uni)serial modules and rings are given. Moreover, the structure of Noetherian left cyclic-uniform uniserial rings are characterized. Finally, we study rings $R$ have the property that every finitely generated $R$-module is cyclic-uniform serial.

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R. Nikandish, M. J. Nikmehr, A. Yassine. 2022-08-16. Cyclic-Uniform Uniserial Modules and Rings. https://arxiv.org/abs/2208.07940

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