Czechoslovak Mathematical Journal, first online, pp. 1-18
On $w$-universal injective modules and their applications in commutative rings
Dechuan Zhou, Hwankoo Kim, Wei Zhao, Kui Hu
Received December 4, 2024. Published online May 5, 2025.
Abstract: Let $R$ be a commutative ring and $w$ be the $w$-operation on $R$. We introduce the concept of $w$-universal injective modules and establish their fundamental properties. It is shown that the product of $E(R/{\frak m})$, where ${\frak m}$ ranges over maximal $w$-ideals of $R$, is a $w$-universal injective $w$-module over $R$, albeit not necessarily a universal injective $R$-module. As applications, we characterize $w$-IF rings and $w$-coherent rings using $w$-universal injective modules. Specifically, we demonstrate that $R$ is a $w$-IF ring if and only if $R$ is $w$-coherent and $E(R/{\frak m})$ is a flat $R$-module for every ${\frak m} \in w \text{-Max}(R)$. These results extend existing results and provide deeper insights into the structure of $w$-modules.
Keywords: $w$-universal injective; $w$-flat module; $w$-IF ring; $w$-coherent ring
Affiliations: Dechuan Zhou, School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, P. R. China, e-mail: zdechuan11119@163.com; Hwankoo Kim (corresponding author), Division of Computer Engineering, Hoseo University, Asan 31499, Republic of Korea, e-mail: hkkim@hoseo.edu; Wei Zhao, Department of Mathematics, Aba Teachers' University, Aba 623002, P. R. China, e-mail: zw9c248@163.com; Kui Hu, School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, P. R. China, e mail: hukui200418@163.com
