Peter Malcolmson

Enveloping algebras of simple three-dimensional Lie algebras

Abstract The Killing form of a simple three-dimensional Lie algebra is shown to classify the Lie algebra and its enveloping algebra. The Lie algebra is isomorphic to the two-by-two trace-zero matrices (the so-called split case) if and only if the form is isotropic in the sense of quadratic form theory. In characteristic zero the skew field of quotients of the enveloping algebra is shown to be isomorphic to a Weyl skew field if and only if the Lie algebra is split. Automorphisms of the Lie algebra and its enveloping algebra are induced by isometries of the Killing form in the nonsplit case.

Determining homomorphisms to skew fields

Given an associative ring R with unit, we may wish to characterize the homomorphisms from R to skew fields (division rings) by means of structure or information defined over R. Then constructing such structure on R will be equivalent to constructing a homomorphism to a skew field. Here we will regard skew field extensions as irrelevant, so that a homomorphism to a skew field composed with an inclusion map into a larger skew field is the same (for our purposes) as the original homomorphism. For example, if R is commutative, a homomorphism from R to a field is determined by the kernel, a prime ideal. The field and the homomorphism may be constructed from the prime ideal by forming the factor ring modulo that ideal and then embedding in the quotient field. For a noncommutative ring, an analog of a prime ideal has been introduced by P. M. Cohn (as in [I]): a “prime matrix ideal” of the ring R. Given a homomorphism R -+ K to a skew field K, we may apply the homomorphism to matrices over R (i.e., with entries in R) by applying the homomorphism to each entry. Then the prime matrix ideal of R determined by R -+ K consists of those square matrices over R whose images under the homomorphism are singular over K. This collection of matrices actually determines the homomorphism (up to isomorphism). In this article we describe several alternative ways of determining homomorphisms to skew fields, including ones induced from the notions of rank of a matrix, linear dependence, and dimension over a skew field. For example, if R ---f K is a homomorphism from a ring to a skew field, one may associate with each finitely presented right R-module A the number dim,(A @ K). This function on finitely presented modules is among the structures shown to determine the homomorphism. The topology that may be put on the collection of such homomorphisms (as on the prime spectrum of a commutative ring) is also discussed in terms of these new determining structures.

Polynomial extensions of IDF-domains and of IDPF-domains

An integral domain is IDF if every non-zero element has only finitely many non-associate irreducible divisors. We investigate when R IDF implies that the ring of polynomials R[T] is IDF. This is true when R is Noetherian and integrally closed, in particular when R is the coordinate ring of a non-singular variety. Some coordinate rings R of singular varieties also give R[T] IDF. Analogous results for the related concept of IDPF are also given. The main result on IDF in this paper states that every countable domain embeds in another countable domain R such that R has no irreducible elements, hence vacuously IDF, and the polynomial ring R[T] is not IDF. This resolves an open question. It is also shown that some subrings R of the ring of Gaussian integers known to be IDPF also have the property that R[T] is not IDPF.

On Fragility of Generalizations of Factoriality

Any class of domains, in particular a class of domains that arises from generalizations of factoriality, invites questions about its stability under the standard operations. One of these generalizations of factoriality is the one that requires that every nonzero element be contained in only finitely many principal prime ideals of height one. We use this property to settle all the open cases in the literature on stability of generalizations of factoriality under the standard ring extensions. The paper provides a compendium on the stability, under ring extensions, of all the known generalizations of factoriality. We also use stability properties of factorization in extensions of valuation domains to give a new characterization of discrete valuation domains.

Rings Without Maximal Ideals

It is traditional in an abstract algebra class to prove, using Zorns lemma, that a ring with unit must have maximal ideals. Without a unit element this is not true, and here we present some commutative counterexamples. First we consider rings with trivial multiplication, i.e., those for which any product is zero. Then an ideal is just an additive subgroup, and we are seeking abelian groups without maximal subgroups. Such groups are easily characterized using the notion of divisibility. If G is an abelian group, written additively, and m is a positive integer, then denote by mG the set {mglg c G}. Then G is said to be divisible if mG = G for every positive integer m. It is easy to verify that the additive group Q of rational numbers is divisible and that every direct sum of divisible groups is divisible.