FIm-modules over Noetherian rings

In this paper we study representation theory of the category ${\mathrm{FI}}^m$ introduced in [6], [7] which is a product of copies of the category FI, and show that quite a few interesting representation theoretic and homological properties of FI can be generalized to ${\mathrm{FI}}^m$ in a natural way. In particular, we prove the representation stability property of finitely generated ${\mathrm{FI}}^m$-modules over fields of characteristic 0.     

Group rings of Noetherian groups

Translated from Matematicheskie Zametki, Vol. 46, No. 6, pp. 61–66, December, 1989.     

Uniserial Noetherian centrally essential rings

Abstract

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings.     

On noncommutative piecewise Noetherian rings

We extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.     

Global dimension of Noetherian serial rings

The global dimension of Noetherian serial rings is studied. It is proved that if an indecomposable serial ring has infinite global dimension then it is Artinian and its quiver is a simple cycle. Using methods of the theory of right serial quivers, we give an upper estimate on the Loewy length of Artinian rings of finite global dimension. Applications to the calculation of the global dimension of tiled orders of width 2 are given.     

On arithmetic Macaulayfication of Noetherian rings

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

     

Finite Modules of Finite Injective Dimension Over a Noetherian Algebra

Let R be a commutative Noetherian ring. Let P(R) (respectively,I(R)) be the category of all finite R-modules of finite projective(respectively, injective) dimension. Sharp [9] constructed acategory equivalence between I(R) and P(R) for certain Cohen–Macaulaylocal rings R. Thus many properties about finite modules offinite projective dimension can be connected with those of finiteinjective dimension through this equivalence.     

Transcendental division algebras and simple Noetherian rings

In this paper we prove the following theorem: Let D be a division ring with center the field k, and let k (x ₁, …, x_n) denote the rational function field in n variables over k. If D contains a maximal subfield which has transcendence degree at least n over k, then D ⊗_k k (x₁, …, x_n) is a simple Noetherian domain of Krull and global dimensions n. Rather surprisingly, the preceding result can be used to determine the maximum transcendence degrees of the commutative subalgebras of several classically studied division rings. Using the theorem we prove, for example, that in the division ring of quotients of the Weyl algebra,A _n, every maximal subfield has transcendence degree at mostn over the center.