Let $R$ be a commutative ring with identity, and $n\geq 1$ an integer. A proper submodule $N$ of an $R$-module $M$ is called an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1, \ldots , a_{n+1}\in R$ and $m\in M$, then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$. In this paper, we study $n$-prime submodules as a generalization of prime submodules. Among other results, it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$, then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$.
@article{33962,
title = {On a generalization of prime submodules of a module over a commutative ring},
journal = {Boletim da Sociedade Paranaense de Matem\'atica},
volume = {37},
year = {2017},
doi = {10.5269/bspm.v37i1.33962},
language = {EN},
url = {http://dml.mathdoc.fr/item/33962}
}
Fazaeli Moghimi, Hosein; Zarei Jalal Abadi, Batool. On a generalization of prime submodules of a module over a commutative ring. Boletim da Sociedade Paranaense de Matemática, Tome 37 (2017) . doi : 10.5269/bspm.v37i1.33962. http://gdmltest.u-ga.fr/item/33962/