A note on trinomial identities in associative algebras

In the paper, trinomial identities of associative algebras over an infinite field are considered (in general, the algebras may have no unit), i.e., identities of the form alpha m(1) + beta m(2) + gamma m(3) = 0, where alpha, beta, and gamma are scalars and m(1), m(2), and m(3) are different monomials. It is shown that any nontrivial identity if this kind implies a semigroup identity. The algebras with trinomial identities are characterized in the language of varieties.