On \(\phi\)-2-absorbing primary submodules
Let \(R\) be a commutative ring with identity and \(M\) be a unitary \(R\)-module. Denote the set of submodules of \(M\) by \(S(M)\). Assume that \(\phi:S(M)\to S(M)\cup\{\emptyset\}\) is a function and \(\phi(N)\subseteq N\) for every submodule \(N\) of \(M\). The authors call a proper submodule \(N\) of \(M\) a \(\phi\)-2-absorbing submodule, when for every \(r,s\in R\) and \(m\in M\) from \(rsm\in N\setminus \phi(N)\) we can deduce that \(rm\in N\) or \(sm\in N\) or \(rs\in \sqrt{(N:M)}\). This concept generalizes \(\phi\)-2-absorbing primary ideals defined in [\textit{A. Badawi} et al., ``Generalizations of 2-absorbing primary ideals of commutative rings'', Turk. J. Math. 40, 703--717 (2016)]. In Section 2, some general results on \(\phi\)-2-absorbing primary submodules are stated. For example, setting \(\phi_i(N)=(N:M)^{i-1}N\) and \(\phi_\omega(N)=\bigcap_{i=1}^\infty \phi_i(N)\), it is proved that if \(n\geq 4\) then a submodule of \(M\) is \(\phi_n\)-2-absorbing primary if and only if it is \(\phi_\omega\)-2-absorbing primary. In Section 3, the behaviour of \(\phi\)-2-absorbing primary submodules under some module constructions, such as taking quotients or localization, is studied. I should mention that I think the definition of \(\phi_T\) on page 33 of the paper is not well-defined. For example, if \(M=R={\mathbb{Z}}\), \(N=2{\mathbb{Z}}\), \(N'=6{\mathbb{Z}}\), \(T=\{3^n|n=0,1,\ldots\}\), \(\phi(N)=\emptyset\) and \(\phi(N')=12{\mathbb{Z}}\), then according to this definition, we get \(\phi_T(N_T)=\emptyset\) and \(\phi_T(N'_T)=12{\mathbb{Z}}_T\), while \(N_T=N'_T\).
scientific article; zbMATH DE number 6693058
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