On graded distributive modules (Q2518000)

The author investigates several characterizations of graded distributive modules over a \({\mathbb Z}\)-graded commutative ring \(R\) with identity. He proves the following two theorems: Theorem \(A\): The following statements are equivalent: (i) \(M\) is a graded distributive \(R\)-module; (ii) Every *closed submodule of \(M\) is *irreducible; (iii) For any \(i\in \mathbb Z\) and \(x,y \in M_i\) and any *maximal ideal \(m\), the graded submodules \(Rx(m)\) and \(Ry(m)\) are comparable with respect to inclusion; (iv) For each graded submodule \(N\) of \(M\) and each maximal ideal \(m\), containing \(N:_R M\), the module *\(E_R(M/N(m))\) is *indecomposable. Theorem \(B\): For a graded \(R\)-module \(M\), the following statements are equivalent:(i) \(M\) is a graded distributive \(R\)-module; (ii) For each graded submodule \(N\) of \(M\), \(N = \bigcap _{p\in mK(M/N)}N(p)\) is *irreducible decomposition of \(N\). Editorial remark: According to the retraction note [Zbl 1441.13002], this article has been retracted because it ``contains a significant amount of overlap with the following article by the same author [``Characterizations of graded distributive modules'', J. Appl. Math. 5, No. 16 (2008), \url{https://www.sid.ir/FileServer/JE/134520081609.pdf}].