The isotopically simple algebras with a nil-basis

Abstract

An algebra is called isotopically simple if every its isotope is a simple algebra. We prove that every two simple algebras are isotopic in the classes of simple 3-dimensional anticommutative or commutative algebras with a nil-basis. For every we give examples of n-dimensional isotopically simple anticommutative (resp. commutative) algebras with a nil-basis. The cases n = 3 and are fundamentally different.Communicated by Pavel Kolesnikov