Ideal and factor conditions for crossed modules and algebra-algebroids

Abstract

In this paper, using the equivalence between the category of crossed modules of algebras and the category of algebra-algebroids, we will explore the notions of ideality and factors for these algebraic structures. We give the structure of a two sided ideal of an algebra-algebroid and the notion of quotient algebra-algebroid. By considering a two sided ideal of an algebra-algebroid, we show that the crossed module corresponding to this ideal is a crossed ideal of the crossed module corresponding to the algebra-algebroid. Conversely, by taking a crossed ideal of a crossed module, we also show that the corresponding algebra-algebroid to this crossed ideal is an ideal of the algebra-algebroid corresponding to the crossed module.

Keywords

• Algebra-Algebroids
• category
• crossed modules

References

1. [1] R. Brown and O. Mucuk, Covering groups of non-connected topological groups revis- ited, Math. Proc. Camb. Philos. Soc. 115, 97–110, 1994.
2. [2] R. Brown and C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indag. Math. (N.S.) 79, 296–302, 1976.
3. [3] G. J. Ellis, Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra, 52 (3), 277–282, 1988.
4. [4] B. Mitchell, Rings with several objects, Advances in Mathematics, 8 (1) 1–161, 1972.
5. [5] B. Mitchell, Some applications of module theory to functor categories, Bull. Amer. Math. Soc. 84, 867–885, 1978.
6. [6] B. Mitchell, Separable Algebroids, Mem. Amer. Math. Soc. 57 (333), 96 pp, 1985.
7. [7] G. H. Mosa, Higher dimensional algebroids and crossed complexes, PhD Thesis, University College of NorthWales, Bangor, 1986.
8. [8] O. Mucuk, T. Şahan and N. Alemdar, Normality and Quotients in Crossed Modules and Group-groupoids. Appl. Categ. Structures 23 (3), 415–428, 2015.

9. [9] K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (2), 129–146, 1990.
10. [10] T. Porter, Homology of commutative algebras and an invariant of Simis and Vascon- celos, J. Algebra, 99, 458–465, 1986.
11. [11] T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. 30, 373–381, 1987.
12. [12] N. M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, PhD Thesis, University College of North Wales, Bangor, 1992.
13. [13] T. Şahan and O. Mucuk, Normality and Quotient in the Category of Crossed Modules Within the Category of Groups with Operations, Bol. Soc. Paran. Mat. 38 (7), 169– 179, 2020.