S-Noetherian properties on amalgamated algebras along an ideal

Let A and B be commutative rings with identity, f:A→B a ring homomorphism and J an ideal of B. Then the subring A{bowtie}^fJ:={(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, we investigate a general concept of the Noetherian property, called the S-Noetherian property which was introduced by Anderson and Dumitrescu, on the ring A{bowtie}^fJ for a multiplicative subset S of A{bowtie}^fJ. As particular cases of the amalgamation, we also devote to study the transfers of the S-Noetherian property to the constructions D+(X₁, . . ., X_n)E[X₁, . . ., X_n] and D+(X₁, . . ., X_n)E{left open bracket}X₁, . . ., X_n{right open bracket} and Nagata's idealization.