Ideals in Arbitrary Three-Dimensional Algebras

TL;DR

This paper classifies all ideals in arbitrary 3-dimensional algebras, providing explicit descriptions and bounds, revealing that such algebras have either finitely many or infinitely many ideals, with specific maximum counts.

Contribution

It offers a complete classification of ideals in 3-dimensional algebras, including explicit descriptions and bounds, for the first time.

Findings

Algebras have either finitely many or infinitely many ideals.

Maximum of 3 one-dimensional ideals per algebra.

Maximum of 2 two-dimensional ideals per algebra.

Abstract

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all ideals of A, providing an explicit characterization of both 1-dimensional and 2-dimensional ideals. Moreover, we classify 2-dimensional ideals into 4 distinct types. We prove that A either has infinitely many ideals or at most 4. We also show that, in any case, the maximum number of 1-dimensional ideals is 3, while the maximum number of 2-dimensional ideals is 2. Finally, we present a class of algebras with a finite number of ideals that attain this theoretical maximum.