Basic module theory

(c) μ(rs,m) = μ(r, μ(s,m)) (d) if 1 ∈ R, then μ(1,m) = m. We shall usually omit the notation of μ and simply write r ·m for μ(r,m). Thus axiom (c) would be written (rs) ·m = r · (s ·m), etc. Exercise 1. We denote by EndGrp(M) the set of group endomorphisms of M : An element φ ∈ EndGrp(M) is a group homomorphism φ : M → M . EndGrp(M) is naturally a ring, with addition and multiplication defined by (φ+ ψ)(m) := φ(m) + ψ(m) and (φψ)(m) := φ ◦ ψ(m). EndGrp(M) has an identity element 1 = idM . Show that the data of a left R-module structure on M is equivalent to giving a (unital, if 1 ∈ R) ring-map R→ EndGrp(M). Remark 1.2. As the name “left R-module” suggests, there is also the notion of a right R-module: A right R-module is an abelian group M together with an external law of composition μ : M × R → M , satisfying the same axioms of Definition 1.1 but with the places of the ring and the abelian group switched. Exercise 2. Can a right R-module structure on M be thought of as a map R→ EndGrp(M)? What conditions must be placed on R or the map R → EndGrp(M) for this to be true? In general, formulate a definition of right R-modules in the spirit of Exercise 1.