Noetherian Algebras Research Articles

A New Measure of Growth for Countable-Dimensional Algebras. I

A new dimension function on countable-dimensional algebras (over a field) is described. Its dimension values lie in the unit interval [0, 1]. Since the free algebra on two generators turns out to have dimension $0$ (although conceivably some Noetherian algebras might have positive dimension!), this dimension function promises to distinguish among algebras of infinite $GK$dimension.

A new measure of growth for countable-dimensional algebras. I

A new dimension function on countable-dimensional algebras (over a field) is described. Its dimension values lie in the unit interval [0, 1]. Since the free algebra on two generators turns out to have dimension 0 0 (although conceivably some Noetherian algebras might have positive dimension!), this dimension function promises to distinguish among algebras of infinite G K GK dimension.

Examples in non-commutative projective geometry

Let A = k ⊕ ⊕n ≥ 1An connected graded, Noetherian algebra over a fixed, central field k (formal definitions will be given in Section 1 but, for the most part, are standard). If A were commutative, then the natural way to study A and its representations would be to pass to the associated projective variety and use the power of projective algebraic geometry. It has become clear over the last few years that the same basic idea is powerful for non-commutative algebras; see, for example, [ATV1, 2], [AV], [Sm], [SS] or [TV] for some of the more significant applications. This suggests that it would be profitable to develop a general theory of ‘non-commutative projective geometry’ and the foundations for such a theory have been laid down in the companion paper [AZ]. The results proved there raise a number of questions and the aim of this paper is to provide negative answers to several of these.

Invariant algebras and completely reducible representations

We give a general construction of affine noetherian algebras with the property that every finite dimensional representation is completely reducible. Starting from enveloping algebras of semi simple Lie algebras in characteristic zero we obtain explicit examples and describe some of their properties.

Gelfand-Kirillov Dimension of Noetherian Semigroup Algebras

It is shown that for certain classes of semigroup algebras K[ S], including right noetherian algebras, the Gelfand-Kirillov dimension is finite whenever it is finite on all cancellative subsemigroups of S. Moreover, the dimension of the algebra modulo the prime radical is then an integer. A description of cancellative semigroups of polynomial growth, extending Gromov′s theorem, has been recently obtained by Grigorchuk. Some bounds on GK( K[ S]) are determined. Our approach is based on the structure of the image S of S modulo the prime radical of K[ S], on the correspondence between the cancellative subsemigroups in S and S and on Grigorchuk′s result.

Affine Noetherian Algebras and Extensions of the Base Field

Affine Noetherian Algebras and Extensions of the Base Field

A new measure of growth for countable-dimensional algebras

A new dimension function on countable-dimensional algebras (over a field) is described. Its dimension values for finitely generated algebras exactly fill the unit interval [0, 1]. Since the free algebra on two generators turns out to have dimension 0 (although conceivably some Noetherian algebras might have positive dimension!), this dimension function promises to distinguish among algebras of infinite GK-dimension.

Flat embeddings of noetherian algebras in artinian rings

It is shown that a noetherian algebraR with finite Gelfand-Kirillov dimension and right primary decomposition can be embedded in an artinian ringS, and thatS is flat as a leftR-module if and only if all right associated primes are minimal. IfR is irreducible then such a flat embedding is possible if and only ifR has an artinian quotient ring. Also, the existence of a left flat embedding in an artinian ring allows an explicit description of the prime middle annihilators ofR.

Behaviour of krull dimension under extension of the base field

Let R be a Noetherian algebra over a field k. A formula is given for the Krull dimension of the ring R⊗k k(X) in terms of the heights of simple modules with large endomorphism rings.

Lattices over Noetherian algebras

An analog of the reduction theorem for modules over an integrally closed integral domain is proved for torsionfree modules over a semiprimary Noetherian algebra. The result is translated into concrete terms for pseudo-Bass and pseudohereditary algebras, and also used to study the structure of genera.

Localization and ideal theory in iterated differential operator rings

In this paper we study primality, hypercentrality, simplicity, and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings. We give sufficient conditions for the skew enveloping algebra of a nilpotent Lie algebra with coefficient ring containing the rational numbers to be a simple ring, and we give necessary and sufficient conditions in the case that the Lie algebra is Abelian. Our main results show that if L is a finite dimensional solvable Lie algebra over a field k of characteristic zero and R is an Artinian or a commutative Noetherian algebra over k, then the skew enveloping algebra R# U( L) satisfies the second layer condition. We discuss consequences of this for localization and use the localization theory to state a classical Krull dimension versus global dimension inequality when k is uncountable.

Embeddings in Artinian rings and Sylvester rank functions

Using Sylvester rank functions it is shown that a right Noetherian algebra, modulo the torsion ideal determined by elements regular modulo its nil radical, is embeddable in a simple Artinian ring. This is used to show that a right Noetherian algebra which is right Krull homogeneous embeds in a simple Artinian ring, and that a right Noetherian affine algebra satisfying a polynomial identity embeds in a simple Artinian ring.

A Generalized Jacobian Criterion for Regularity

For a commutative noetherian algebra $B$ over a perfect field $A$ regularity is equivalent to the flatness of ${\Omega _{B|A}}$ plus ${H_1}(A,B,B) = 0$ (simplicial homology). In characteristic 0 the homological condition is superfluous.

Gelfand-Kirillov Dimension is Exact for Noetherian PI Algebras

AbstractIf O → A → C → B → O is a short exact sequence of finitely generated modules over a Noetherian Pi-algebra then we show that GK(C) = max{GK(A), GK(B)}.

P-rings and P-homomorphisms

All rings considered in this paper are unitary. commutative and noetherian: the notations are those of N. Bourbaki, A. Grothendieck and H. Matsumura; in particular, for a semi-local ring A, 2 means the radical adic completion of A. Let R be a property of a local noetherian ring and P the following property (associated to R) of a noetherian algebra A over a field k IS, 7.3 I: “The k-algebra A satisfies P if and on1.v $ for any finite extension k’ of k. the local rings of A mj), k’ sati@ the property R”. A ring homomorphism q~: A --f R is called a P-homomorphism if it is flat and if, for any prime ideal p of A, the k(p)-algebra B 0, k(p) satisfies P: by a P-ring, we mean a ring A such that, for any prime ideal p of A, the canonical homomorphism A p ---) a p is a P-homomorphism. This paper is devoted, for a nice property R, to the following problems asked by A. Grothendieck (8, 7.4, 7.5 I:

Localisation and completions of Noetherian PI algebras

Localisation and completions of Noetherian PI algebras

The number of generators of a module over a fully bounded ring

We prove a theorem giving an upper bound for the number of generators of a module in terms of local data, for a class of rings which includes Noetherian rings which are right fully bounded. Forster and Swan have proved such results for commutative Noetherian rings and finite algebras over these rings. Eisenbud and Evans have shown, further, that from any given set of generators for a finitely generated module over such an algebra, the number predicted by the theorem can be obtained by a certain kind of elementary transformation. This result we also obtain for our wider class of rings. Regarding these results as results about the existence and structure of epimorphisms F -+ A, where F is free and A the module, we obtain also corresponding results were F is a direct sum of copies of some projective module other than R. The chief difficulties throughout are those involved in finding substitutes for arguments involving localization, which is not an available tool in our context. For prime rings of Krull dimension one, we can remove the boundedness condition, and also obtain for hereditary Noetherian prime (HNP) rings the analogue of a result of Roiter’s for lattices over orders. We also obtain a weaker result for epimorphisms F + A, where F is a direct sum of copies of some finitely presented module which is not projective, and a uniqueness result (up to right equivalence) for epimorphisms F + A, where F is a large enough free module. In the following, terms like “Noetherian” and “Goldie”-&e applied on both sides unless specified otherwise. If A is a module, we let the direct sum of n copies of A be denoted wi (rather than A”), in or&r to avoid confusion when referring to ideals. If A is a module, we let g(A) be the smallest number which is the cardinality of a set of generators for A. A prime ring is right bounded if every essential right ideal contains a nonzero two-sided ideal, and a ring R is right fully bounded if for every prime ideal P, R/P is right

Krull dimension of noetherian algebras and extensions of the base field

Krull dimension of noetherian algebras and extensions of the base field

Localization of fully left bounded rings

This paper generalizes properties which hold for localization of Azumaya algebras, in two directions. Firstly, fully left bounded left Noetherian rings, especially finitely generated Noetherian algebras, are considered. It is noted that for such rings every idempotent kernel functor a is symmetric, i.e. the filter T(σ) of a-dense left ideals has a basis of a-dense ideals. A prime ideal P of a f.l.b.l.N. ring R is localizable if and only if it is the intersection of the P-critical left ideals. In case R is a finitely generated algebra over its (Noetherian) center C, we apply the technique of “descent” of kernel functors. If a is a symmetric kernel functor such that R(A n c) S T(σ) for every A G T(σ) and such that a has property (T) then there is a kernel functor a’ on C-modules such that Qσ (R) ≅Q≅ ,(R). If P is a prime ideal of R then σ- descends to C if and only if P is localizable. Secondly, a class of rings is described in terms of the Zariski topology on Spec. The imposed condition is weaker than maxi...