Free modules and crossed modules of R-algebroids

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 introducethe Peiffer ideal of a precrossed module and use it to construct the free crossed module

___

  • [1] Alp M. Pullback crossed modules of algebroids. Iran J Sci Tech; 2008; 32: 1-5.
  • [2] Alp M. Pushout crossed modules of algebroids. Iran J Sci Tech; 2008; 32: 175-181.
  • [3] Amgott SM. Separable categories. J Pure Appl Algebra 1986; 40: 1-14.
  • [4] Arvasi̇Z. Crossed modules of algebras. Mathematical and Computational Applications 2004; 9: 173-182.
  • [5] Arvasi Z, Porter T. Simplicial and crossed resolutions of commutative algebras. J Algebra 1996; 181: 426-448.
  • [6] Baues HJ, Conduché D. The central series for Peiffer commutators in groups with operators. J Algebra 1990; 133: 1-34.
  • [7] Brown R, Higgins PJ, Sivera R. Nonabelian Algebraic Topology. Helsinki, Finland: European Mathematical Society, EMS Tracts in Mathematics Vol. 15, 2011.
  • [8] Brown R. Homotopies and automorphisms of crossed modules of groupoids. Appl Categor Struct 2003; 11: 185-206.
  • [9] Brown R, Huebschmann J. Identities among relations. Low-dimensional topology, Bangor, 1979. Lond Math S 1982; 48: 153-202.
  • [10] Ellis GJ. Higher dimensional crossed modules of algebras. J Pure Appl Algebra 1988; 52: 277-282.
  • [11] Gerstenhaber M. On the deformation of rings and algebras: II. Ann Math 1966; 84: 1-19.
  • [12] Kassell C, Loday JL. Extensions centrale d’algèbres de Lie. Ann Inst Fourier 1982; 32: 119-142 (in French).
  • [13] Lichtenbaum S, Schlessinger M. The cotangent complex of a morphism. T Am Math Soc 1967; 128: 41-70.
  • [14] Lue AST. Non-abelian cohomology of associative algebras. Q J Math 1968; 19: 159-180.
  • [15] Mitchell B. Rings with several objects. Adv Math 1972; 8: 1-161.
  • [16] Mitchell B. Some applications of module theory to functor categories. B Am Math Soc 1978; 84: 867-885.
  • [17] Mitchell B. Separable Algebroids. Providence, RI, USA: AMS, 1985.
  • [18] Mosa GH. Higher dimensional algebroids and crossed complexes. PhD, University of Wales, Cardiff, UK, 1986.
  • [19] Porter T. Homology of commutative algebras and an invariant of Simis and Vasconcelos. J Algebra 1986; 99: 458-465.
  • [20] Porter T. Some categorical results in the theory of crossed modules in commutative algebras. J Algebra 1987; 109: 415-429.
  • [21] Porter T. The Crossed Menagerie: An introduction to crossed gadgetry and cohomology in algebra and topology. 17 April 2018. https://ncatlab.org/nlab/files/menagerie12a.pdf.
  • [22] Shammu NM. Algebraic and categorical structure of categories of crossed modules of algebras. PhD, University of Wales, Cardiff, UK, 1992.
  • [23] Whitehead JHC. On adding relations to homotopy groups. Ann Math 1941; 42: 409-428.
  • [24] Whitehead JHC. Note on a previous paper entitled “On adding relations to homotopy groups.”. Ann Math 1946; 47: 806-810.