Throughout the paper, R denotes a commutative Noetherian ring with nonzero unity and I and J denote arbitrary ideals of R. The amalgamated of R along an ideal J, which was introduced and studied in [2, 5, 6] (see, also [3, 4] for its natural generalization), denoted by R J, is defined as the following subring of R × R: R x J = {(r, r + j); r ∈ R, j ∈ J}. This construction has several applications in difference contexts (see, for example, [7, 13]). Clearly, by using the scalars product s(r, r + j) = (sr, sr + sj), for all s ∈ R, and (r, r + j) ∈ R x J, it follows that R x J has an R-module structure.

Keywords