Showing papers on "Free algebra published in 1979"

Definable principal congruences in varieties of groups and rings

Abstract: In [lJ Baldwin and Berman showed that for varieties ~ with DPC (definable principal congruences) certain results of Taylor concerning residually small varieties could be sharpened. Their question as to whether every variety generated by a finite algebra has DPC was answered in the negative in [2]; however the question remained open for varieties with permutable congruences. The study of DPC became even more interesting when McKenzie [4] proved that this property could be used, in certain cases (such as a variety generated by a para-primal algebra), to give an easy proof of the finite axiomatizablity of the variety. McKenzie then showed that among lattices only the distributive varieties have DPC, and states that the question of whether varieties generated by a finite group or ring have DPC is open. In the first section we point out that a variety ~ has DPC iff the free algebra on countably many generators in ~ has SDPC (strongly definable principal congruences), hence a variety generated by a class Y{ of algebras has DPC iff the quasi-variety generated by 5g has DPC. In the second section a finite ring R is constructed such that the variety generated by R does not have DPC. In the third section we prove that if the variety generated by a finite group G has DPC then G must be nilpotent; on the other hand if G is nilpotent class 2 and finite then indeed it generates a variety with DPC. It follows that the properties of having DPC and being finitely axiomatizable are independent for quasi-varieties generated by a finite group. Finally Baldwin's theorem (3) that the variety of all groups of exponent 3 has DPC is shown to be best possible for Burnside varieties.

Polynomial functions over monoids

Abstract: In this paper we study the algebra of polynomial functions over a monoid (S,.,e). We consider the congruences and a natural binary relation on this algebra of polynomial functions in several variables.

A PI-algebra which is not PI when an inverse is adjoined

Abstract: An example is produced of an algebra, embedded in 2 x 2 matrices over a field, which is not PI when an element's inverse is formally adjoined. This example is used to show that the generic 2 x 2 matrices, a domain, has the same property. In commutative algebra the procedure of adjoining the inverse of an element in a ring has proven useful. Recently work has been done towards generalizing the concepts of commutative algebra and algebraic geometry to PI-algebras [1], [2], [6]. In this paper it is shown that the algebra which is obtained from a subring of a ring of 2 x 2 matrices over an extension of the base field, by formal adjunction of the inverse of an element is not PI. We do this by describing an irreducible infinite dimensional representation and applying Kaplansky's theorem [4]. For A and B algebras over a commutative ring C, we shall use the notation of the free product A *c B as found in [5]. This C-algebra is the coproduct in the category of C-algebras. For R an algebra over a field k, and x an element of R, let x' denote the residue class of z in the algebra R *k k[z]/(zx 1, xz 1). Define R{x'} to be the above algebra. Let k{X, Y} be the algebra obtained by adjoining two generic 2 x 2 matrices to a field k. THEOREM 1. k{X, Y} {X'} is not a PI-algebra. To prove this theorem we shall work with another algebra. We define A to be the quotient of the free algebra k{x, y} by the relations y2 = 0 and yx'y = 0, for i > 0. LEMMA 2. The set S = {xiyxJ, xk; ij,k > 0) spans A as a vector space over k. PROOF. The set S u {0} is closed under multiplication, since x vxJxkyxt = 0, for ij,k,n > 0. Thus since A is generated as an algebra by x andy, the ring A is spanned by S over k. Note. S is actually a basis for A. Received by the editors July 25, 1978 and, in revised form, October 18, 1978. AMS (MOS) subject classifications (1970). Primary 16A08; Secondary 16A38.

The Countable Chain Condition and Free Algebras

Abstract: Let A be an algebra that contains a constant 0 and a binary operation 9 among its fundamental operations Elements x, yeA are called disjoint if x y = 0 A satisfies the countable chain condition (ccc) if it contains no uncountable set of pairwise disjoint nonzero elements Consider the following facts: 1 (Classical) A free Boolean algebra satisfies ccc 2 (Abian [-1]) A free algebra in the ring variety generated by a finite field satisfies ccc These examples suggest the question ~ ccc, hold in every free algebra of a quasi-primal variety?" Before this question can be answered we must know what the ccc, means in a universal algebra In Section 2 we propose a general algebraic condition, called the countable separability condition (csc), and show that, in many situations, it is equivalent to ccc Using csc, we give an affirmative answer to the posed question (Theorem 29) In Section 1 the free algebras in quasi-primal varieties are determined The main tool for their description is the second duality result for quasi-primal varieties due to Keimel and Werner [2] For unexplained notation and terminology see [2]