On the semiprimitivity of finitely generated algebras

Finitely generated associative algebras A = K〈a1, . . . , an〉 over a field K defined by homogeneous relations are considered. If there exists an order on the associated free monoid FMn of rank n such that the set of normal forms of elements of A is a regular language in FMn, then the algebra A is semiprimitive provided that the associated monomial algebra is semiprime. Automaton algebras were defined by Ufnarovskii by the condition that the set of normal forms of elements of the algebra is a regular language; see [16]. More precisely, if A is a finitely generated algebra over a field K with a set of generators a1, . . . , an, then let K〈x1, . . . , xn〉 denote the free K-algebra of rank n and let π : K〈x1, . . . , xn〉 −→ A be the homomorphism such that π(xi) = ai, for i = 1, . . . , n. Assume that a well order≺ is given on the free monoid FMn = 〈x1, . . . , xn〉 such that the unity of FMn is the least element and that is compatible with the multiplication in FMn. The latter means that v ≺ u implies that vw ≺ uw and wv ≺ wu for all u, v, w ∈ FMn. Let I = Iπ be the ideal of FMn consisting of all leading monomials of elements of ker(π). Then the set N(A) = FMn \I is called the set of normal words corresponding to the chosen presentation of the algebra A and the chosen order on FMn. The minimal set of generators of I is called the set of obstructions. The image K[FMn]/K[I] of the free algebra K[FMn] is referred to as the monomial algebra associated to A and is an important tool in the combinatorial aspects of finitely generated algebras; see [6, 16]. One says that A is an automaton algebra if N(A) is a regular language; see [16, Section 5.10]. Recall that this means that this set is obtained from a finite subset of FMn by applying a finite sequence of operations of union, multiplication and operation ∗ defined by T ∗ = ⋃ i≥1 T , for T ⊆ FMn. It is well known that N(A) is a regular language if and only if the set of obstructions is a regular language [16, Theorem 5.2.2]. Therefore, the class of automaton algebras contains the class of algebras with a finite set of obstructions. In other words, if K[FMn]/K[I] is a finitely presented algebra, then A is an automaton algebra. Consequently, due to the definition of Gröbner bases, if an algebra A admits a finite Gröbner basis, then A is an automaton algebra; see [16, §2]. For basic results on regular languages and automata theory we refer the reader to [8]. Received by the editors April 11, 2012 and, in revised form, February 1, 2013. 2010 Mathematics Subject Classification. Primary 16S15, 16N20; Secondary 16S36, 20M25, 68Q70. This work was supported by MNiSW research grant N201 420539. c ©2014 American Mathematical Society Reverts to public domain 28 years from publication