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Free modules and crossed modules of R-algebroids

In this paper, first, we construct the free modules and precrossed modules of R-algebroids. Then we introduce the Peiffer ideal of a precrossed module and use it to construct the free crossed module. | Turk J Math (2018) 42: 2863 – 2875 © TÜBİTAK doi:10.3906/mat-1601-38 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Free modules and crossed modules of R-algebroids 1 Osman AVCIOĞLU1,∗,, İbrahim İlker AKÇA2 , Department of Mathematics, Faculty of Arts and Science, Uşak University, Uşak, Turkey 2 Department of Mathematics and Computer Sciences, Faculty of Science and Letters, Eskişehir Osmangazi University, Eskişehir, Turkey Received: 11.01.2016 • Accepted/Published Online: 11.04.2016 • Final Version: 27.11.2018 Abstract: In this paper, first, we construct the free modules and precrossed modules of R -algebroids. Then we introduce the Peiffer ideal of a precrossed module and use it to construct the free crossed module. Key words: R-category, R-algebroid, crossed modules, free modules 1. Introduction Crossed modules, algebraic models of two types, were first invented by Whitehead [23, 24] in his study on homotopy groups and have been studied by many mathematicians. Various studies on crossed modules over groups and groupoids can be found in papers and books such as [7, 8, 21], and those over algebras in [4, 5, 19, 20, 22] and in [11, 13, 14] in different names. Kassell and Loday [12] studied crossed modules of Lie algebras and higher dimensional analogues were proposed by Ellis [10] for use in homotopical and homological algebras. Mosa [18] studied crossed modules of R -algebroids and double algebroids. Pullback and pushout crossed modules of algebroids can be found in [1] and [2], respectively. Provided that P is a group and K is a set, the construction of the free P -group on K and the constructions of the free precrossed and crossed modules on a function ω : K −→ P were handled in [7]. Shammu constructed the free crossed module on a function f : K −→ A where, with our notations, K is a set and A is an R -algebra for a commutative ring R in [22]. The basic goal of this paper is to construct the free R -algebroid .