Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study the basic theory of the category of graded $w$-Noetherian modules over a graded ring $R$. Some elementary concepts, such as $w$-envelope of graded modules, graded $w$-Noetherian rings and so on, are introduced. It is shown that:(1) A graded domain $R$ is graded $w$-Noetherian if and only if $R_\fkm^g$ is a graded Noetherian ring for any gr-maximal $w$-ideal $\fkm$ of $R$, and there are only finite numbers of gr-maximal $w$-ideals including $a$ for any nonzero homogeneous element $a$. (2) Let $R$ be a strongly graded ring. Then $R$ is a graded $w$-Noetherian ring if and only if $R_e$ is a $w$-Noetherian ring. (3) Let $R$ be a graded $w$-Noetherian domain and let $a\in R$ be a homogeneous element. Suppose $\fkp$ is a minimal graded prime ideal of $(a)$. Then the graded height of the graded prime ideal $\fkp$ is at most $1$.

Keywords: Graded $w$-envelope of a module, graded $w$-exact sequence, graded finitely presented type module, graded $w$-Noetherian ring

MSC numbers: 13C13, 13E99

Supported by: This work was partially supported by the National Natural Science Foundation of China (11671283, 11661014 and 11961050)
and the Guangxi Natural Science Foundation (2016GXSFDA380017)