On the Ideal Transforms Defined by an Ideal

Abstract Let $$R$$ be a commutative Noetherian ring, $$I$$ an ideal of $$R$$ , and $$M$$ an $$R$$ -module. The ambiguous structure of $$I$$ -transform functor $$D_{I}(-)$$ makes the study of its properties attractive. In this paper we gather conditions under which $$D_{I}(R)$$ and $$D_{I}(M)$$ appear in certain roles. It is shown under these conditions that $$D_{I}(R)$$ is a Cohen–Macaulay ring, regular ring, $$\cdots$$ and $$D_{I}(M)$$ can be regarded as a Noetherian, flat, $$\cdots R$$ -module. We also present a primary decomposition of zero submodule of $$D_{I}(M)$$ and secondary representation of $$D_{I}(M)$$ .