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⋈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⋈fJ for a multiplicative subset S of A⋈fJ. As particular cases of the amalgamation, we also devote to study the transfers of the S-Noetherian property to the constructions D+(X1,…,Xn)E[X1,…,Xn] and D+(X1,…,Xn)E〚X1,…,Xn〛 and Nagataʼs idealization.