Research Article
BibTex RIS Cite
Year 2023, Volume: 52 Issue: 3, 698 - 707, 30.05.2023
https://doi.org/10.15672/hujms.1131802

Abstract

References

  • [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] 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] G. J. Ellis, Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra, 52 (3), 277–282, 1988.
  • [4] B. Mitchell, Rings with several objects, Advances in Mathematics, 8 (1) 1–161, 1972.
  • [5] B. Mitchell, Some applications of module theory to functor categories, Bull. Amer. Math. Soc. 84, 867–885, 1978.
  • [6] B. Mitchell, Separable Algebroids, Mem. Amer. Math. Soc. 57 (333), 96 pp, 1985.
  • [7] G. H. Mosa, Higher dimensional algebroids and crossed complexes, PhD Thesis, University College of NorthWales, Bangor, 1986.
  • [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] K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (2), 129–146, 1990.
  • [10] T. Porter, Homology of commutative algebras and an invariant of Simis and Vascon- celos, J. Algebra, 99, 458–465, 1986.
  • [11] T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. 30, 373–381, 1987.
  • [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] 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.

Ideal and factor conditions for crossed modules and algebra-algebroids

Year 2023, Volume: 52 Issue: 3, 698 - 707, 30.05.2023
https://doi.org/10.15672/hujms.1131802

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.

References

  • [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] 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] G. J. Ellis, Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra, 52 (3), 277–282, 1988.
  • [4] B. Mitchell, Rings with several objects, Advances in Mathematics, 8 (1) 1–161, 1972.
  • [5] B. Mitchell, Some applications of module theory to functor categories, Bull. Amer. Math. Soc. 84, 867–885, 1978.
  • [6] B. Mitchell, Separable Algebroids, Mem. Amer. Math. Soc. 57 (333), 96 pp, 1985.
  • [7] G. H. Mosa, Higher dimensional algebroids and crossed complexes, PhD Thesis, University College of NorthWales, Bangor, 1986.
  • [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] K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (2), 129–146, 1990.
  • [10] T. Porter, Homology of commutative algebras and an invariant of Simis and Vascon- celos, J. Algebra, 99, 458–465, 1986.
  • [11] T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. 30, 373–381, 1987.
  • [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] 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.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Özgün Gürmen Alansal 0000-0003-2851-986X

Publication Date May 30, 2023
Published in Issue Year 2023 Volume: 52 Issue: 3

Cite

APA Gürmen Alansal, Ö. (2023). Ideal and factor conditions for crossed modules and algebra-algebroids. Hacettepe Journal of Mathematics and Statistics, 52(3), 698-707. https://doi.org/10.15672/hujms.1131802
AMA Gürmen Alansal Ö. Ideal and factor conditions for crossed modules and algebra-algebroids. Hacettepe Journal of Mathematics and Statistics. May 2023;52(3):698-707. doi:10.15672/hujms.1131802
Chicago Gürmen Alansal, Özgün. “Ideal and Factor Conditions for Crossed Modules and Algebra-Algebroids”. Hacettepe Journal of Mathematics and Statistics 52, no. 3 (May 2023): 698-707. https://doi.org/10.15672/hujms.1131802.
EndNote Gürmen Alansal Ö (May 1, 2023) Ideal and factor conditions for crossed modules and algebra-algebroids. Hacettepe Journal of Mathematics and Statistics 52 3 698–707.
IEEE Ö. Gürmen Alansal, “Ideal and factor conditions for crossed modules and algebra-algebroids”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 3, pp. 698–707, 2023, doi: 10.15672/hujms.1131802.
ISNAD Gürmen Alansal, Özgün. “Ideal and Factor Conditions for Crossed Modules and Algebra-Algebroids”. Hacettepe Journal of Mathematics and Statistics 52/3 (May 2023), 698-707. https://doi.org/10.15672/hujms.1131802.
JAMA Gürmen Alansal Ö. Ideal and factor conditions for crossed modules and algebra-algebroids. Hacettepe Journal of Mathematics and Statistics. 2023;52:698–707.
MLA Gürmen Alansal, Özgün. “Ideal and Factor Conditions for Crossed Modules and Algebra-Algebroids”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 3, 2023, pp. 698-07, doi:10.15672/hujms.1131802.
Vancouver Gürmen Alansal Ö. Ideal and factor conditions for crossed modules and algebra-algebroids. Hacettepe Journal of Mathematics and Statistics. 2023;52(3):698-707.