# On generalisations of almost prime and weakly prime ideals

Document Type : Research Paper

Author

‎University ‎of Vali-e-Asr

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 $a_1,\ldots,a_n\in R$‎, ‎$a_1\cdots a_n\in P\backslash P^m$ ($a_1\cdots a_n\in P\backslash \{0\}$) implies $a_1\cdots a_{i-1}a_{i+1}\cdots a_n\in P$‎, ‎for some $i\in\{1,\ldots,n\}$; ($m,n\geq 2$)‎. ‎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.

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### History

• Receive Date: 16 November 2012
• Revise Date: 07 April 2013
• Accept Date: 12 April 2013
• First Publish Date: 01 April 2014