Endomorphism Ring of 𝑮𝒁-Regular Modules

Abstract

As a complement to the former work of 𝐺𝑍-regular modules, in this paper along the lines of 𝑍- regular modules due to Zelmanowitz, we improve the study of the endomorphism ring of 𝑍- regular modules to 𝐺𝑍-regular modules. We give a sufficient condition on 𝐺𝑍-regular module 𝑀 such that 𝑆 = 𝐸𝑛𝑑(𝑀) is 𝜋-regular ring and we prove that 𝑅-module 𝑀 is 𝐺𝑍-regular if and only if 𝑆 = 𝐸𝑛𝑑(𝑀) is 𝜋 -regular ring in case that 𝑀 is a projective finitely power generated 𝑅-module. Also we show that for a 𝐺𝑍-regular 𝑅-module 𝑀, the center of 𝑆 = 𝐸𝑛𝑑(𝑀) , 𝐶𝑒𝑛(𝑆) , is 𝜋 -regular ring. Even further if 𝑀 is a 𝐺𝑍 -regular 𝑅 -module then 𝑅/𝑎𝑛𝑛(𝑀) is dense in 𝐶𝑒𝑛(𝑆).