Ideals in arbitrary three-dimensional algebras

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis together with its multiplication table, which is specified by 27 structure constants. We describe all ideals of A , providing an explicit characterization of both one-dimensional and two-dimensional ideals. Moreover, we analyze all possible configurations of ideals in a three-dimensional algebra and provide examples illustrating each possible case. More precisely, we prove that the number of non-trivial ideals of a 3-dimensional algebra is either infinite or at most 6, with the exception of 5.

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