Strongly sum 2-irreducible submodules of a module
A non-zero submodule \(N\) of a module \(M\) over a commutative ring \(R\) is called sum 2-irreducible if whenever it is the sum of three submodules of \(M\), it is already the sum of two of the three. This notion was introduced in [\textit{F. Farshadifar} and \textit{H. Ansari-Toroghy}, Jordan J. Math. Stat. 14, No. 3, 545--554 (2021; Zbl 1488.13035)]. In the present paper, the authors define and study the concept of strongly sum 2-irreducibility. A non-zero submodule \(N\) of \(M\) is strongly sum 2-irreducible if whenever it is contained in the sum of three submodules of \(M\), it is already contained in the sum of two of the three. The main result, Theorem 1, characterizes strongly sum 2-irreducible submodules in terms of completely irreducible submodules: submodules of \(M\) that cannot be written as the intersection of a family of strictly larger submodules of \(M\). Characterizations of strongly sum 2-irreducibility are provided for submodules \(N\) having certain additional properties, and for submodules of \(M\) in the case that \(R\) is a finite product of constituent rings.
scientific article; zbMATH DE number 7626125
scientific article