부산대학교 도서관

Let $G$ be a group and let $R$ be a commutative $G$-graded ring withunity 1. Recall that a proper graded ideal $P$ of $R$ is said to begraded $r$-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 $b^n\in P$ forsome $n\in \Bbb N$.\parIn this article, the authors introduce graded uniformly $pr$-ideals ofa graded commutative ring and prove some relations between this classand graded uniformly primary ideals, graded strongly primary ideals andgraded $r$-ideals. More precisely, they prove that the class of graded$pr$-ideals contains the class of graded uniformly $pr$-ideals and thatthe class of graded uniformly $pr$-ideals contains the class of gradedstrongly $pr$-ideals. Also, they prove that graded $pr$-ideals, gradeduniformly $pr$-ideals and graded strongly $pr$-ideals are equivalent inany $\Bbb Z$-graded Noetherian ring.