CLASSES OF DUALLY SLENDER MODULES

The natural categorial notion of a compact object, for which the covariant functor Hom commutes with direct sums, was first time studied in the context of module theory in 60’s. Hyman Bass gave a non-categorial characterization of the notion (see Lemma 1.1 of the present paper) in the book [3] and basic properties of such a module were published by Rentschler in [10]. The notion has been studied under various terms (module of type Σ, small, Σ-compact, U-compact module), we use the term dually slender following the terminology of [6]. The study of dually slender modules has been motivated by progress of research in various branches of algebra. Probably the most frequent motivation (and the closest to the author of the present paper) comes from the context of representable equivalences of module categories ([4], [5], [13], [14] etc.). Dually slender modules has appeared also in the structure theory of graded rings [9] or almost free modules [12]. The structure theory of dually slender modules was developed also in [6],[18], [15]. The present paper has an expository and survey character, however it contains several new results (Lemma 2.1, Proposition 2.7, Proposition 3.5) which generalize and simplify older concepts. The first section, which introduces the central notions and their basic properties, is followed by an exposition of functorial properties of classes of dually slender modules and their consequences. The last part is devoted to a description of the structure of classes of dually slender modules over particular types of rings. The results cited in the paper are mostly published in [18], [11], [15], [16] and [17]. Throughout the paper a ring R means an associative ring with unit, and a module means a right R-module. We will use the letter R for a ring in all claims. The minimal cardinality of a set of generators of an R-module M is denoted by genR(M). A ring R is (von Neumann) regular provided that each x ∈ R has a pseudo-inverse element, i.e. there is y ∈ R satisfying xyx = x. A regular ring R is abelian regular if all idempotents of R are central. We refer for non-explained terminology to [1].