부산대학교 도서관

Let $G$ be a group and let $R$ be a commutative $G$-graded ring with unity 1. Recall that a proper graded ideal $P$ of $R$ is said to be graded $r$-ideal if whenever $a, b \in h(R)$ such that $ab \in P$ with $Ann(a) = \{0\}$, then $b \in P$. Here, $h(R)$ is equal to $\bigcup_{g\in G}R_g$. The proper ideal $P$ is called a graded $pr$-ideal if $ab \in P$ with $Ann(a) = \{0\}$ gives $b^n\in P$ for some $n\in \Bbb N$. \par In this article, the authors introduce graded uniformly $pr$-ideals of a graded commutative ring and prove some relations between this class and graded uniformly primary ideals, graded strongly primary ideals and graded $r$-ideals. More precisely, they prove that the class of graded $pr$-ideals contains the class of graded uniformly $pr$-ideals and that the class of graded uniformly $pr$-ideals contains the class of graded strongly $pr$-ideals. Also, they prove that graded $pr$-ideals, graded uniformly $pr$-ideals and graded strongly $pr$-ideals are equivalent in any $\Bbb Z$-graded Noetherian ring.