A Frattini theory for evolution algebras

TL;DR

This paper introduces a Frattini theory for evolution algebras, defining subalgebras and ideals, and explores their properties and implications in the structure of these algebras.

Contribution

It develops a new Frattini theory for evolution algebras, including definitions, properties, and conditions for triviality, extending classical algebra concepts to this setting.

Findings

Defined the Frattini subalgebra and ideal for evolution algebras.

Provided necessary and sufficient conditions for their triviality.

Explored the role of the Frattini ideal in dually atomistic evolution algebras.

Abstract

This paper develops a Frattini theory for evolution algebras defining the Frattini subalgebra as the intersection of all maximal subalgebras, and the Frattini ideal as the largest ideal contained in it. To this end, we revisit the notion of nilradical, whose classical definition is not directly applicable in this setting, and propose the supersolvable nilradical as a suitable alternative. This leads to necessary and sufficient conditions for the triviality of the Frattini subalgebra and ideal. Finally, we also briefly examine the relevance of the Frattini ideal in the study of dually atomistic evolution algebras.