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.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data:We give the structure of a two sided ideal of an algebra-algebroid and the notion of quotient algebra-algebroid.
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Conclusion & Impact: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.
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.
Alansal, Ö. G. (2023). Ideal and factor conditions for crossed modules and algebra-algebroids. Hacettepe Journal of Mathematics and Statistics, Volume: 52 Issue: 3. https://doi.org/10.15672/hujms.1131802
G. H. Mosa, Higher dimensional algebroids and crossed complexes, PhD Thesis, University College of NorthWales, Bangor, 1986.
O. Mucuk, T. Şahan and N. Alemdar, Normality and Quotients in Crossed Modules and Group-groupoids. Appl. Categ. Structures 23 (3), 415–428, 2015.
K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (2), 129–146, 1990.
T. Porter, Homology of commutative algebras and an invariant of Simis and Vascon- celos, J. Algebra, 99, 458–465, 1986.
T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. 30, 373–381, 1987.
N. M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, PhD Thesis, University College of North Wales, Bangor, 1992.
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.