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Czechoslovak Mathematical Journal, first online, pp. 1-19

The essential submodules of incidence modules

Hailou YAO, Naseer ULLAH, Qianqian YUAN

Received May 19, 2025.   Published online September 21, 2026.
Abstract:  Let $R$ be a ring with an identity, $M$ be a left $R$-module, and $I(X,M)$ be the incidence module of a locally finite partially ordered set $X$ over $M$. We give necessary and sufficient conditions for the existence of the minimum essential submodules in $I(X,M)$. We also compute the socle and the singular submodule of the incidence module $I(X,M)$ for some $X$ in terms of the socle of $M$ and the singular submodule of $M$, respectively.
Keywords:  incidence module; incidence ring; essential submodule; singular submodule; socle
Classification MSC:  13C13, 16D70, 16D80, 16D99
DOI:  10.21136/CMJ.2026.0183-25
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Affiliations:   Hailou Yao, Naseer Ullah, School of Mathematics, Statistics and Mechanics, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing 100124, P. R. China, e-mail: yaohl@bjut.edu.cn, naseerullah530@gmail.com; Qianqian Yuan (corresponding author), Department of Basic Sciences, Beijing International Studies University, Beijing 100024, P. R. China, e-mail: cherryyuan77@126.com
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