Søren Jøndrup

Automorphisms and derivations of upper triangular matrix rings

Abstract Kezlan proved that for a commutative ring C , every C -automorphism of the ring of upper triangular matrices over C is inner. We generalize this result to rings in which all idempotents are central; moreover we show that for a semiprime ring A and central subring C , every C -automorphism of the ring of upper triangular matrices over C is the composite of an inner automorphism and an automorphism induced from a C -automorphism of A . By the method of proof we re-prove results of S. P. Coelho and C. P. Milies and of Mathis, stating that a derivation of a ring of upper triangular matrices of a C -algebra ( n × n matrices over A ) is a sum of an inner derivation and a derivation induced from a C -derivation of A . By an example we show that an extra assumption is needed for proving the above result of automorphisms of upper triangular matrices. Finally we consider automorphisms of subrings of n × n matrices over a commutative ring C , where entries over the diagonal are from C and below the diagonal are taken from a nil ideal. We prove that all such automorphisms are inner.

Representations of Some PI Algebras

Abstract In this paper we find the degree of quantizations of Weyl algebras. Moreover in some cases we describe representations of maximal degree. For the quantized Weyl algebras, we prove that certain nice localizations are quasipolynomial algebras. Finally we use the general result to get the centers of the various algebras hence generalizing and reproving results from Awami et al. (1988) and Morikawa (1989). Essential for many of our results is a description of the simple R[Θ; α]-modules in terms of the k-automorphism α and the simple R-modules, here R is a prime affine PI algebra, k an algebraically closed field of characteristic 0 and α a k-automorphism of finite order.

Centres and fixed-point rings of artinian rings

It is well known that the centre of a left artinian ring R is usually not artinian; however considering the centre Z(R) as the endomorphism ring of the (R | R~ R of finite length, it follows that Z(R) is necessarily semiprimary, i.e. the Jacobson radical J(R) is nilpotent and R/J(R) is artinian. It is the purpose of this note to describe (partially) the commutative semiprimary rings which can appear as centres of artinian rings. In Section 1 we show that if a commutative ring R is the fixed-point ring for a finitely generated group of automorphisms of an artinian ring, then R is also centre of an artinian ring. In Section 2 we firstly prove that any commutative semiprimary ring, for which the square of the Jacobson radical vanishes, can be embedded in an artinian ring. Next we see that many of these semiprimary rings actually appear as centres of artinian rings. Here the assumption that the square of the Jacobson radical is zero, is essential as shown by Example 2.1. In this paper a ring will always mean an associative ring with an identity element, subrings have the same identity element and homomorphisms preserve the identity element. J(R) denotes the Jacobson radical of the ring R.

The Geometry of Noncommutative Plane Curves

We consider (noncommutative) affine algebras over algebraically closed fields. A 1-dimensional representations of such an algebra, A, corresponds to a point on an algebraic set in an affine space. Different points of (nonisomorphic 1-dimensional simple A-modules) can have nonzero “Ext”-groups. We will show these groups will in many cases give information back to the algebra. In this note, we mainly focus on algebras of the form k ⟨ x, y ⟩/(f). We show that if such an algebra has a representation x → a and y → b for all points P = (a, b) in the plane and nonzero “Ext” for all such, then f ∈ ([x, y])2. Another case we can solve completely. If an algebra A = k ⟨ x, y ⟩/(f), where f ∈ ([x, y]) has for all points in the plane, then f = [x, y] + f 0, where f 0 ∈ ([x, y])2. We show by examples that the 1-dimensional representations and their corresponding “Ext”-groups can be the same, but the higher dimensional simple representations are in general quite different. We also prove that each curve has a model with an n-dimensional simple representation for any n > 1.

Towards a more general notion of Gelfand-Kirillov dimension

Gelfand-Kirillov dimension (GK) has proved to be a useful invariant for algebras over fields. In this paper we generalize the notion of GK to algebras over commutative Noetherian rings by replacing vector space dimension with reduced rank. It turns out that most results about GK have analogues for the new GK.

Slender domains and compact domains

Abstract We prove that a one-dimensional Noetherian domain is slender if and only if it is not a local complete ring. The latter condition for a general Noetherian domain characterizes the domains that are not algebraically compact. For a general Noetherian domain R we prove that R is algebraically compact if and only if R satisfies a condition slightly stronger than not being slender. In addition we enlarge considerably the number of classes of rings for which the question of slenderness can be answered. For instance we prove that any domain, not a field, essentially of finite type over a field is slender.

DIMENSION OF NONCOMMUTATIVE PLANE CURVES

In this paper, we prove that an algebra of the form k〈x,y〉/(f) is never right (or left) artinian in case (f) is a proper ideal and k is an uncountable, algebraically closed field of characteristic 0.

SIMPLE MODULES AND NON-COMMUTATIVE QUADRICS

In this paper we classify non-commutative quadrics and study their homological properties. In fact we find all non-commutative algebras of degree 2 up to isomorphism and we study these algebras via their homomorphic images onto the polynomial algebra k[x,y] as well as the Ext1(k(p), k(q))-groups, where k(p) and k(q) are one-dimensional simple modules. Moreover some general results on simple finite-dimensional modules are obtained. Some of these results are applied to the special cases of non-commutative quadrics.

Some remarks on growth of algebras over commutative noetherian rings

Using a growth function,GK defined for algebras over integral domains, we construct a generalization of Gelfand Kirillov dimensionGGK. GGK coincides with the classical no-tion of GK for algebras over a field, but is defined for algebras over arbitrary commutative rings. It is proved that GGK exceeds the Krull dimension for affine Noetherian PI algebras. The main result is that algebras of GGK at most one are PI for a large class of commutative Noetherian base rings including the ring of integers, Z. This extends the well-known result of Small, Stafford, and Warfield found in [11].