Varieties satisfying semigroup identities: algebras over a finite field and rings (Q2821825)

With any associative algebra \((A,+, \cdot)\) over a commutative and associative ring with 1, two semigroups are associated in a natural way. The first one is just the multiplicative semigroup \((A,\cdot)\) of the algebra. The second one is a so-called adjoint semigroup \((A, \circ)\), where the multiplication \(\circ\) (circle composition) is defined by letting \(a\circ b = a + b - ab\) for all \(a, b \in A\). A polynomial identity \(u = v\) is said to be a semigroup identity, if \(u, v \) are distinct words written with the multiplication \(\cdot\), and an adjoint semigroup identity, if \(u,v\) are distinct words written with the operation \(\circ\).NEWLINENEWLINEIn this article, several varieties of associative algebras over a finite field and varieties of associative rings satisfying semigroup or adjoint semigroup identities are studied. The author characterizes these varieties in terms of ``forbidden algebras'' (that is, of algebras from the given list that do not belong to such varieties) and discusses some corollaries of the characterizations.