About Theory of Primary Decomposition of Monomial Ideal

In this paper, we have to R is a commutative Noetherian ring, i.e. where all ideal is finitely generated, and we have the R-module I(G), which is a monomial ideal, where I(G) is the edge ideal of a simple and finite graph G, with no isolated vertices, which is a finitely generated R-module. We consider also α an ideal of R and N a submodule of I(G) such that α I (G)⊆N, an inclusion of modules together with the edge ideal. Here in the article, the edge primary decomposition and irreducible decomposition of
α x N are given.