On Injective Modules and Related Topics

Abstract

The main objective of this work is to study injective modules and related topics. Due to the relations between injective and divisible modules, a notion of several properties about both modules was studied. We introduce all of the key definitions used in the thesis, as well as some findings about the injective module a pear. Every injective module is divisible, but the inverse requires an additional condition P.I.D. Also, if the ring R is semi simple, and M is a semi simple R-module, then M is injective. Also, if M is a cyclic and the regular module is injective. Also, if M is regular with N≤ M is a finitely generated submodule, so M is injective. Here, we study several relationships between injective modules over the Dedekind domain. We prove that every divisible module M over D.V.R is injective. Also, any pseudo injective module with R is a Dedekind leads to M is an injective. Several facts about the relationship between the injective module and the Euclidean ring are satisfied here. And we find that every divisible module on the Noetherian valuation ring is injective. Also, there are some connections between (D.V.R) and the injective module. Finally, we investigate the injective module in relation to other rings, such as the Noetherian ring, the local ring, the D.V.R, and the hereditary ring. We prove that if M is an R-module and I is a maximal ideal of R with quotient ring R_I of R is a field, then M is an injective module. Also, if R is a D.V.R, so any I-divisible module over R is injective.