We consider algebras over a field K with a presentation K<x1,…,xn:R>, where R consists of [Formula: see text] square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. The description of all four generated algebras of this type that satisfy a certain non-degeneracy condition is given. The structure of one of these algebras is described in detail. In particular, we prove that the Gelfand–Kirillov dimension is one while the algebra is noetherian PI and semiprime in case when the field K has characteristic zero. All minimal prime ideals of the algebra are described. It is also shown that the underlying monoid is a semilattice of cancellative semigroups and its structure is described. For any positive integer m, we construct non-degenerate algebras of the considered type on 4m generators that have Gelfand–Kirillov dimension one and are semiprime noetherian PI algebras.
Prime affine algebras of Gelfand—Kirillov dimension one
