Abstract. Let A and B be commutative rings with unity, f : A → B a ring homomorphism and J an ideal of B. Then the subring A ⊲⊳ f J := {(a, f (a) + j)|a ∈ A and j ∈ J} of A × B is called the amalgamation of A with B along with J with respect to f . In this paper, among other things, we investigate the Cohen-Macaulay and (quasi-)Gorenstein properties on the ring A ⊲⊳ f J.

Let A and B be commutative rings with unity, f : A → B a ring homomorphism and J an ideal of B. Then the subring A ⊲⊳ f J := {(a, f (a) + j)|a ∈ A and j ∈ J} of A × B is called the amalgamation of A with B along J with respect to f . In this paper, we study the property of Cohen-Macaulay in the sense of ideals which was introduced by Asgharzadeh and Tousi, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring A ⊲⊳ f J. Among other things, we obtain a generalization of the well-known result that when the Nagata's idealization is Cohen-Macaulay.2010 Mathematics Subject Classification. 13A15, 13C14, 13C15.

In this paper we improve the recent results on the transfer of Prüfer, Gaussian and arithmetical conditions on amalgamated constructions. As an application we provide an answer to a question posed by Chhiti, Jarrar, Kabbaj and Mahdou as well as we construct various examples.2010 Mathematics Subject Classification. Primary 13F05, 13A15.Proof. Assume that (r, f (r) + j) ∈ Z(R ⊲⊳ f J). Then (r, f (r) + j)(s, f (s) + j ′ ) = 0 for some non-zero element (s, f (s) + j ′ ) ∈ R ⊲⊳ f J. Hence rs = 0 and jf (s) + j ′ (f (r) + j) = 0. If s = 0, then j ′ = 0 and j ′ (f (r) + j) = 0. Otherwise, r ∈ Z(R). This proves the inclusion.We already have the inclusion {(r, f (r) + j) | j ′ (f (r) + j) = 0, for some j ′ ∈ J \ {0}} ⊆ Z(R ⊲⊳ f J). To complete the proof, it is enough for us to show that {(r, f (r) + j) | r ∈ Z(R) and j ∈ J} ⊆ Z(R ⊲⊳ f J) under the validity of any of

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The purpose of this paper is to study certain categorical properties of the categories SoftAct of all soft S-acts and soft homomorphisms, and WSoftAct of all soft S-acts and weak soft homomorphisms. We investigate the interrelations of some particular morphisms, limits and colimits in SoftAct and WSoftAct with their corresponding notions in the categories ActS and Set. It is proved that SoftAct has non-empty soft coproducts and soft coequalizers and then soft pushouts. Moreover, WSoftAct has arbitrary w-soft products and non-empty w-soft coproducts. Some results concerning soft equalizers and w-soft pullbacks are also presented.