p-clean properties in amalgamated rings

Abstract

Let A be a ring. Then A is called p-clean ring if each element in A express as the sum of an idempotent and pure element. Let f : A → B be a ring homomorphism and J be an ideal of B. The amalgamation of A with B along J with respct to f is a new ring structure introduced and studied by Anna et al. in 2009. This construction is a generalization of the amalgamated duplication of a ring along an ideal and other classical constructions such as the A + XB[X] and A + XB[[X]] constuctions. In this paper, the transfer of the notion of p-clean rings to the amalgamation of rings along ideal is studied. In particular, the necessary and sufficient conditions for amalgamation to be a p-clean ring are studied.

Keywords

• pure element, amalgamation ring, von neumann ring