Delta operation on modules, prime and radical submodules and primary decomposition

Year 2020, Volume: 49 Issue: 4, 1303 - 1314, 06.08.2020

Ashkan Nikseresht

https://doi.org/10.15672/hujms.667410

Abstract

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. In this paper, in order to study prime submodules, radical submodules and primary decompositions in finitely generated free $R$-modules, we introduce and study an operation $\Delta: (M\oplus R)^2\to M$ defined by $\Delta(m+r, m'+r')= r'm-rm'$. In particular, using this operation we give a characterization of prime submodules of $M\oplus R$, in terms of prime submodules of $M$. As an application, we present a characterization of prime submodules of finitely generated free modules. Also we present a formula for the prime radical of submodules of $M\dis R$. Moreover, we state some conditions under which primary decompositions of submodules of $M$ lift to $M\oplus R$.

Keywords

Delta operation, primary decomposition, prime submodule, radical of submodules

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