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Semilattices of algebras and the description of radicals of semigroup algebras

In the use of semigroup algebras for studying semigroup representations and in the study of the structure of semigroup algebras, it is important to know how their radicals are constructed. In the work of Amitsur, Weisglas, Wauters, Gilmer, Jespers, Karpilovsky, Krempa, Mann, Okninskii, Puschyelovskii, Sierpinski, and others, various aspects of the problem of the description of semigroup algebras KS are considered, where S is a commutative semigroup and Κ is an associative algebra over a commutative ring Φ with an identity. A survey of this approach is given in [1]. In this note we state results completely describing the Jacobson and Brown —McCoy radicals of semigroup algebras of commutative semigroups, as well as semigroups satisfying at least one identity with the rewriting property. We recall that an identity is said to have the rewriting property if it has the form xj ... xn = xal ... xan, where σ is a non-trivial permutation. In the study of semigroup algebras, the notion of semilattices of algebras, introduced by Weisglas in 1973, proves to be useful. Let Ω be a semilattice, that is, a commutative semigroup of idempotents. An algebra R is called the semilattice Ω of all its subalgebras Λα, α e Ω, if the Φ-module Λ is a direct sum of the Φ-modules Ra and R^Rp £ Ra$ for any α, β in Ω. The algebras Ra are called the components