Bimultiplications and Annihilators of Crossed Modules in Associative Algebras

Year 2021, Issue: 35, 72 - 90, 30.06.2021

Ummahan Ege Arslan , Serdar Hürmetli

https://doi.org/10.53570/jnt.905298

Cited By: 2

Abstract

In this paper, we present a generalization of the concept of the bimultiplication algebra by defining the bimultiplication of crossed modules in associative algebras. Using this structure, we construct the actor of a crossed module in associative algebras, and we obtain annihilators of crossed modules as the kernel of a crossed module morphism from a crossed module to its actor. Moreover, we construct a link between this morphism and a crossed square, two-dimensional crossed module.

Keywords

Bimultiplication crossed module, actor crossed module, annihilator of crossed module, crossed square

References

• S. M. Lane, Extensions and Obstructions for Rings, Illinois Journal of Mathematics 121 (1958) 316-345.
• J. H. C. Whitehead, Combinatorial Homotopy I, Bulletin of the American Mathematical Society 55 (1949) 231-245.
• J. H. C. Whitehead, Combinatorial Homotopy II, Bulletin of the American Mathematical Society 55 (1949) 453-456.
• S. Lichtenbaum, M. Schlessinger, The Cotangent Complex of a Morphism, Transactions of the American Mathematical Society 128(1) (1967) 41-70.
• M. Gerstenhaber, On the Deformation of Rings and Algebras: II, Annals of Mathematics 84(1) (1966) 1-19.
• P. Dedecker, A. S. T. Lue, A Nonabelian Two-Dimensional Cohomology for Associative Algebras, Bulletin of the American Mathematical Society 72(6) (1966) 1044-1050. K. J. Norrie, Actions and Automorphisms of Crossed Modules, Bulletin de la Société Mathématique de France 118 (1990) 129-146.
• A. S. T. Lue, Semi-Complete Crossed Modules and Holomorp hs of Groups, Bulletin of the London Mathematical Society 11 (1979) 8-16.
• Z. Arvasi, U. Ege, Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures 11 (2003) 487-506.
• S. Hürmetli, Bimultiplication of Crossed Modules, PhD Dissertation, Eskişehir Osmangazi University, (2016) Eskişehir, Turkey.
• Y. Boyacı, J. M. Casas, T. Datuashvili, E. Ö. Uslu, Actions in Modified Categories of Interest with Application Crossed Modules, Theory and Applications of Categories 30(25) (2015) 882-908.
• G. J. Ellis, Higher Dimensional Crossed Modules of Algebras, Journal of Pure and Applied Algebra 52(3) (1988) 277-282.
• A. S. -T. Lue, Non-Abelian cohomology of Associative Algebras, The Quarterly Journal of Mathematics 19(1) (1968) 159-180.
• R. Lavendhomme, Th. Lucas, On Modules and Crossed Modules, Journal of Algebra 179 (1996) 936-963.
• Z. Arvasi. Crossed Squares and 2-Crossed Modules of Commutative Algebras, Theory and Applications of Categories 3(7) (1997) 160-181.
• Z. Arvasi, E. Ulualan, Homotopical Aspects of Commutative Algebras I: Freeness Conditions for Crossed Squares, Journal of Homotopy and Related Structure 10 (2013) 495-518.
• E. S. Yılmaz, K. Yılmaz, On Crossed Squares of Commutative Algebras, Mathematical Sciences and Applications E-Notes (8) 2 (2020) 32-41.