On Minimal Approximations of Modules

Let R be a ring and consider the category ModR of (right) R-modules. Given a class C ofR-modules, a morphismM → N in ModR is called a left C-approximation of M provided that N belongs to C and the induced morphism HomR(N,C) → HomR(M,C) is surjective for every C in C. This concept has been introduced by Auslander and Smalø [2], and independently by Enochs [5]. Approximations arise naturally in various situations, for instance in representation theory of finite dimensional algebras, in particular in tilting theory, or in commutative algebra. In this note we address the problem of finding minimal approximations. In general, a left C-approximation M → N of M is not uniquely determined by M , but it happens quite often that there exists an approximation φ : M → N which is left minimal in the sense that any endomorphism ψ of N satisfying ψ ◦φ = φ is an isomorphism. Note that such a minimal approximation of M is unique up to isomorphism. This paper is divided into four sections. In Section 1 we characterize the morphisms φ : M → N which admit a decomposition φ = (φ, φ) : M → N ′ ∐ N ′′ = N such that φ is left minimal and φ = 0. For example, such a decomposition always exists if N is pure-injective. In Section 2 we discuss criteria for a class C of R-modules such that every R-module admits a left C-approximation. In Section 3 we introduce a new class of modules M which we call product-complete since the class AddM of direct summands of coproducts of copies of M is closed under taking products. We prove that an R-moduleM is product-complete if and only if every R-module admits a (minimal) left AddM-approximation. In fact, we provide a host of characterizations of such modules which indicate that productcomplete modules are particulary well-behaved. For example, every endofinite module is product-complete. More precisely, we prove in Section 4 that a module M is endofinite if and only if every direct summand of M is product-complete.