Cancellation Modules and Homogeneous Idealization

Abstract

All rings are commutative with identity and all modules are unital. In this article, we characterize cancellation modules and use this characterization to give necessary and sufficient conditions for the sum and intersection of cancellation modules to be cancellation. We introduce and give some properties of the concept of 1 2 join principal submodules. We show that via the method of idealization most questions concerning 1 2 (weak) cancellation and 1 2 join principal modules can be reduced to the ideal case.

Idealization properties of comultiplication modules

“…Conversely, if I is multiplication and N multiplication such annI + [IM : N ] = R, then I (+) N is multiplication. See, for example, [4] Propositions 5 and 7, [3] Theorem 9 and [2] Theorem 9. If I (+) N is idempotent in M , then I is idempotent in R and N is idempotent in M by [9] Theorem 17.…”