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

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 $\Delta: insert ignore into journalissuearticles values(M\oplus R);^2\to M$ defined by $\Deltainsert ignore into journalissuearticles values(m+r, m`+r`);= r`m-rm`$. 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 $M\oplus R$.