Perspectivity of Modules and Some Additive Decompositions of Elements in Rings

Abstract

Cancellation properties and additive decompositions of elements in rings have been extensively studied since long. There are strong inter-relations between some cancellation properties and additive decompositions of elements in rings. A ring R is said to have the internal cancellation property if any isomorphic direct summands of RR have isomorphic direct complements. It was proved by Camillo and Khurana that a von Neumann regular ring R has the internal cancellation property if and only if its every element a can be written as a = e + u, where e is an idempotent and u is a unit, with aR \ eR = 0. Also it has been recently proved by Grover et al. that if a von Neumann regular ring has the internal cancellation property, then its every element can be written as a sum of two units if identity is a sum of two units. We de ne a ring R to be perspective if any two isomorphic direct summands of RR have a common direct complement. It is clear that perspectivity in rings is a stronger property than the internal cancellation property. It turns out that perspectivity in some rings leads to better results about additive decompositions of elements. The goal of this dissertation is to study perspective rings in detail and also some additive decompositions of elements in rings.