On Some Finitely Based Representations Of Semigroups

In this paper we present a method of obtaining finitely based linear representations of possibly infinitely based semigroups. Let R{x1, x2, . . . } be a free associative algebra over a commutative ring R with the countable set of free generators {x1, x2, . . . }. An endomorphism α of R{x1, x2, . . . } is called a semigroup endomorphism if x1α, x2α, . . . are monomials (i.e. finite products of xi’s). An ideal I of R{x1, x2, . . . } is called an S-ideal if I is closed under all semigroup endomorphisms of R{x1, x2, . . . }. Let S be a semigroup, M a faithful module over R. A multiplicative homomorhism ψ : S −→ EndRM is called a (linear) representation of S on M . An element p = p(x1, x2, . . . , xn) of R{x1, x2, . . . } is called an identity of the representation ψ if p(ψ(s1), . . . , ψ(sn)) = 0 for all s1, . . . , sn ∈ S. It is not hard to show that the set of all identities of any representation is an Sideal of R{x1, x2, . . . }. An S-ideal I of R{x1, x2, . . . } is called finitely S-generated if there exist p1, . . . , pk ∈ I such that I is the least S-ideal of R{x1, x2, . . . } containing p1, . . . , pk. A representation ψ is called finitely based if the S-ideal I of its identities is finitely S-generated. Any set {p1, . . . , pk} ⊆ I such that the elements p1, . . . , pk S-generate I is called a finite basis of identities of ψ. S.M.Vovsi and N.H.Shon in [VSh] proved that every representation of a finite group over a field is finitely based. A semigroup version of this problem is open: it is not known whether every representation of a finite semigroup is finitely based. Let A be an associative algebra over R, S = 〈A, ·〉 its multiplicative semigroup, and M be an R-module. Let φ : A −→ EndRM be a homomorphism (R-linear, additive, and multiplicative). The kernel of φ, Kerφ, is a two-sided ideal in A. Define a representation ψ of S on M by mψ(s) = mφ(s) for all m ∈ M , s ∈ S. The representation ψ is a linear representation of S on M . Call the representation ψ associated with φ. From now on we assume that the ring R and the module M have at least one of the following properties: (i) for any integer m there exist r1, . . . , rm ∈ R such that ∏ i<j(ri − rj)m = 0 implies m = 0 for each m ∈ M ; (ii) the ring R is finitely generated. In this paper we prove the following theorem. Received by the editors June 17, 1996 and, in revised form, November 19, 1996. 1991 Mathematics Subject Classification. Primary 20M89, 16R50.