Roughness in modules B. Davvaz * , M. Mahdavipour Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran Received 15 February 2004; received in revised form 31 January 2006; accepted 3 February 2006 Abstract The theory of rough sets was proposed by Pawlak in 1982. The relations between rough sets and algebraic systems have been already considered by many mathemati- cians. Important algebraic structures are groups, rings and modules. Rough groups and rough rings have been investigated by Biswas and Nanda, Kuroki and Wang, and Davvaz. In this paper, we consider an R-module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an R-module, which is an extended notion of a submodule in an R-module. We also give some properties of the lower and the upper approximations in an R-module. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Rough sets; Lower approximation; Upper approximation; R-module; Submodule; Rough submodule 1. Introduction The theory of rough sets was proposed by Pawlak [26] in 1982. The theory of rough sets is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. 0020-0255/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2006.02.014 * Corresponding author. E-mail address: davvaz@yazduni.ac.ir (B. Davvaz). Information Sciences 176 (2006) 3658–3674 www.elsevier.com/locate/ins