Ring of Endomorphisms and Modules over a Ring

Abstract

Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing a module via a ring of endomorphism of an abelian group and the set of all homomorphisms of modules form a module [1] along with Ch. 2 set. 1 of [2]. The formalized items are shown in the below list with notations: Mab for an Abelian group with a suffix “ ab ” and M without a suffix is used for left modules over a ring. 1. The endomorphism ring of an abelian group denoted by End(Mab ). 2. A pair of an Abelian group Mab and a ring homomorphism R → ρ R\mathop \to \limits^\rho End (Mab ) determines a left R-module, formalized as a function AbGrLMod(Mab, ρ) in the article. 3. The set of all functions from M to N form R-module and denoted by Func_Mod R (M, N). 4. The set R-module homomorphisms of M to N, denoted by Hom R (M, N), forms R-module. 5. A formal proof of Hom R (¯R, M) ≅M is given, where the ¯R denotes the regular representation of R, i.e. we regard R itself as a left R-module. 6. A formal proof of AbGrLMod(M′ ab, ρ′) ≅ M where M′ ab is an abelian group obtained by removing the scalar multiplication from M, and ρ′ is obtained by currying the scalar multiplication of M. The removal of the multiplication from M has been done by the forgettable functor defined as AbGr in the article.