A criterion for reflexivity of modules

Let M be a finitely generated module over a ring Λ. With certain mild assumptions on Λ, it is proven that M is a reflexive Λ-module, once M ∼= M∗∗ as a Λ-module. Let Λ be a ring. For each left Λ-module X , let X = HomΛ(X,Λ) denote the Λ-dual of X . This note aims at reporting the following. Proposition 1. Let Λ be a ring and let M be a finitely generated left Λ-module. Assume that one of the following conditions is satisfied. (1) Λ is a left Noetherian ring. (2) Λ is a semi-local ring, that is Λ/J(Λ) is semi-simple, where J(Λ) denotes the Jacobson radical of Λ. (3) Λ is a module-finite algebra over a commutative ring R. Then, M is a reflexive Λ-module, that is the canonical map M hM → M is an isomorphism if and only if there is at least one isomorphism M ∼= M of Λ-modules. To show the above assertion, we need the following. This is well-known, and the proof is standard. Lemma 2. Let N be a right Λ-module and set M = N. Then the composite of the homomorphisms M hM → M (hN ) ∗ → M equals the identity 1M . Proof of Proposition 1. We have only to show the if part. Thanks to Lemma 2, we have a split exact sequence 0 → M hM → M → X → 0 of left Λ-modules, so that M ∼= M ⊕X , since M ∼= M. Therefore, we get a surjective homomorphism ε : M → M with Ker ε = X . Hence, if Condition (1) is satisfied, then X = (0), so that M is a reflexive Λ-module. If Condition (2) is satisfied, then, setting J = J(Λ), we get Λ/J ⊗Λ M ∼= (Λ/J ⊗Λ M)⊕ (Λ/J ⊗Λ X) whence Λ/J ⊗Λ X = (0) by Krull-Schmidt’s theorem, so that X = (0). Suppose that Condition (3) is satisfied. Then, M ∼= M ⊕ X as an R-module, where M is finitely 2020 Mathematics Subject Classification. 13C13, 13D30, 13C12.