Ebrahimpour, M. (2014). On generalisations of almost prime and weakly prime ideals. Bulletin of the Iranian Mathematical Society, 40(2), 531-540.

Mahdieh Ebrahimpour. "On generalisations of almost prime and weakly prime ideals". Bulletin of the Iranian Mathematical Society, 40, 2, 2014, 531-540.

Ebrahimpour, M. (2014). 'On generalisations of almost prime and weakly prime ideals', Bulletin of the Iranian Mathematical Society, 40(2), pp. 531-540.

Ebrahimpour, M. On generalisations of almost prime and weakly prime ideals. Bulletin of the Iranian Mathematical Society, 2014; 40(2): 531-540.

On generalisations of almost prime and weakly prime ideals

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.