Abstract. Let I and J be two ideals of a commutative Noetherian ring R and M be an R-module. For a non-negative integer n it is shown that, if the sets Ass R (Ext

Let I and J be two ideals of a commutative Noetherian ring R and M be an R -module of dimension d . For each i ∈ N0 let H i I,J (−) denote the i -th right derived functor of ΓI,J (−) , where ΓI,J (M ) := {x ∈ M : I n x ⊆ Jx for n ≫ 1} . If R is a complete local ring and M is finite, then attached prime ideals of H d−1 I,J (M ) are computed by means of the concept of co-localization. Moreover, we illustrate the attached prime ideals of H t I,J (M ) on a nonlocal ring R , for t = dim M and t = cd (I, J, M ) , where cd (I, J, M ) is the last nonvanishing level of H i I,J (M ) .

In this paper, we introduce the concept of grey [Formula: see text]-acts and morphisms between grey [Formula: see text]-acts on monoids, which construct a category, namely, [Formula: see text]. Next, we define indecomposable, cyclic, free and projective grey [Formula: see text]-acts. We show that any grey [Formula: see text]-act is a free grey [Formula: see text]-act if and only if it is a free object in this category. Also, we show that any grey [Formula: see text]-act is epimorphism image of any free grey [Formula: see text]-act. We prove that any free grey [Formula: see text]-act is a projective grey [Formula: see text]-act and any cyclic grey [Formula: see text]-act is an indecomposable grey [Formula: see text]-act.

In this paper, we define measurable [Formula: see text]-acts on monoids and construct a new category, namely Meas Act-[Formula: see text]. We investigate the behavior of this category with respect to product, coproduct, pushout, pullback, equalizer and coequalizer. Next, we study the injectivity and complete injectivity in category Meas Act-[Formula: see text] and show any measurable [Formula: see text]-act can be embedded into complete injectivity. Moreover, we investigate the Skornjakov’ theorem for measurable S-acts.