On the decomposition of perfect arbitrary algebras as a direct sum of indecomposable ideals

Antonio J. Calderón Martín; Antonio J. Calderón Martín

doi:10.3934/math.2026611

AIMS Mathematics

2026, Volume 11, Issue 5: 14840-14856. doi: 10.3934/math.2026611

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On the decomposition of perfect arbitrary algebras as a direct sum of indecomposable ideals

Antonio J. Calderón Martín ^,

Department of Mathematics, Faculty of Sciences, University of Cádiz, Campus de Puerto Real, 11510, Puerto Real, Cádiz, Spain

Received: 04 March 2026 Revised: 13 May 2026 Accepted: 19 May 2026 Published: 27 May 2026
MSC : 15A21, 17A01, 17A60

We consider perfect algebras $ A $ (that is, $ A^2 = A $), in their greatest generality. That is, of arbitrary dimension, over an arbitrary base field, and no identity or condition are assumed for the product. We introduce a certain action involving the linear group of $ A $ and provide a characterization and a necessary condition for $ A $, in terms of the above action, to be a direct sum of indecomposable ideals.
- arbitrary algebra,
- decomposition,
- indecomposable ideal,
- perfect algebra,
- linear group,
- structure theory
Citation: Antonio J. Calderón Martín. On the decomposition of perfect arbitrary algebras as a direct sum of indecomposable ideals[J]. AIMS Mathematics, 2026, 11(5): 14840-14856. doi: 10.3934/math.2026611

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Abstract

We consider perfect algebras $ A $ (that is, $ A^2 = A $), in their greatest generality. That is, of arbitrary dimension, over an arbitrary base field, and no identity or condition are assumed for the product. We introduce a certain action involving the linear group of $ A $ and provide a characterization and a necessary condition for $ A $, in terms of the above action, to be a direct sum of indecomposable ideals.

References

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