Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] is a superficial ideal for [Formula: see text] if the following conditions are satisfied: (i) [Formula: see text], where [Formula: see text] denotes a minimal set of generators of an ideal [Formula: see text]. (ii) [Formula: see text] for all positive integers [Formula: see text]. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

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In this paper we investigate some results of multiplication modules on arithmetical rings. We introduce the concept of quasi-copure submodule of an R-module M and we will give some properties of them.

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