On linear difference equations over rings and modules

We develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary com-mutative ground rings. 1. Introduction. Although the theory of linear difference equations over base fields is well understood, the theory over arbitrary ground rings and modules is still under development. It is becoming more interesting and is gaining increasingly special importance mainly because of recent applications in coding theory and cryptography (e.g., [10, 15]). In a series of papers, Taft et al. (e.g., [17, 22, 24]) developed a coalgebraic aspect for the study of linearly recursive sequences over fields. Moreover, Grünenfelder et al. studied in [8, 9] the linearly recursive sequences over finite-dimensional vector spaces. Linearly recursive (bi)sequences over arbitrary rings and modules were studied intensively by Nechaev et al. (e.g., [16, 20, 21]); however, the coalgebraic approach in their work was limited to the field case. Generalization to the case of arbitrary commuta-tive ground rings was studied by several authors including Kurakin [12, 13, 14] and eventually Abuhlail, Gómez-Torrecillas, and Wisbauer [4]. In this paper, we develop a coalgebraic aspect for the study of solutions of linear difference equations over arbitrary rings and modules. For some of our results, we assume that the ground ring is Artinian. Besides the new results, this paper extends results in [4] and " Kapitel 4 " of my doctoral thesis [2]. A standard reference for the theory of linearly recursive sequences over rings and modules is the comprehensive work of Mikhalev et al. [16]. For the theory of Hopf algebras, the reader may refer to any of the classical references (e.g., [1, 19, 23]). By R we denote a commutative ring with 1 R ≠ 0 R and with U(R) = {r ∈ R | r is invertible} the group of units of R. The category of R-(bi)modules will be denoted by ᏹ R. For an R-module M, we call an R-submodule K ⊂ M pure (in the sense of Cohn) if