Properties of Amalgamated Algebras Along an Ideal and Their Applications

If given rings A and B, a ring homomorphism f : A --> B, and an ideal J of B, then a new ring can be constructed called amalgamated algebras along an ideal which is denoted by $A \bowtie^f J := {(a, f(a)+j) \mid a \in A, j \in J}$ with component-wise addition and multiplication. This paper discusses the construction, definition, properties such as isomporhisms, and characterization of amalgamated algebras along an ideal that is a prime ring and a Noetherian ring with detailed explanations. We also discuss its characterization as a reduced ring, which is a continuation from the previous paper. Furthermore, we investigate its characterization as an Artinian ring by adding an additional condition that every ideal of J has unity.