S-small and S-essential submodules
S Rajaee - arXiv preprint arXiv:2109.00519, 2021 - arxiv.org
arXiv preprint arXiv:2109.00519, 2021•arxiv.org
This paper is concerned with S-co-m modules which are a generalization of co-m modules.
In section 2, we introduce the S-small and S-essential submodules of a unitary $ R $-module
$ M $ over a commutative ring $ R $ with $1\neq 0$ such that S is a multiplicatively closed
subset of $ R $. We prove that if $ M $ is an S-co-m module satisfying the S-DAC and $ N\leq
M $, then $ N\leq^{S} _ {e} M $ if and only if there exists $ I\ll^{S} R $ such that $ s (0: _ {M}
I)\leq N\leq (0: _ {M} I) $ for some $ s\in S $. Let $ M $ be a faithful S-strong co-m $ R …
In section 2, we introduce the S-small and S-essential submodules of a unitary $ R $-module
$ M $ over a commutative ring $ R $ with $1\neq 0$ such that S is a multiplicatively closed
subset of $ R $. We prove that if $ M $ is an S-co-m module satisfying the S-DAC and $ N\leq
M $, then $ N\leq^{S} _ {e} M $ if and only if there exists $ I\ll^{S} R $ such that $ s (0: _ {M}
I)\leq N\leq (0: _ {M} I) $ for some $ s\in S $. Let $ M $ be a faithful S-strong co-m $ R …
This paper is concerned with S-co-m modules which are a generalization of co-m modules. In section 2, we introduce the S-small and S-essential submodules of a unitary -module over a commutative ring with such that S is a multiplicatively closed subset of . We prove that if is an S-co-m module satisfying the S-DAC and , then if and only if there exists such that for some . Let be a faithful S-strong co-m -module. We prove that if then there exists an ideal such that . The converse is true if and is a prime module. In section 3, we introduce the S-quasi-copure submodules of an -module and investigate some results related to this class of submodules.
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