On generalisations of almost prime and weakly prime ideals

Abstract

Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a<n> $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-weakly prime ideals are proved‎. ‎We show that in a Noetherian domain a $Phi_m$-prime ideal is primary and we show that in some well known rings $(n-1,n)$-$Phi_m$-prime ideals and $(n-1,n)$-prime ideals coincide‎.