Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Noetherian down-up algebras

Down-up algebras were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that is equivalent to being right (or left) Noetherian, and also to being a domain. Furthermore, when this occurs, we show that is Auslander-regular and has global dimension 3.

     

Witt rings of division algebras over global fields

We study basic properties of Auslander-Gorenstein rings related to duality, localization and purity of minimal injective resolutions.     

Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Let be a differentiably simple Noetherian commutative ring of characteristic (then is local with ). A short proof is given of the Theorem of Harper (1961) on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there exists a nilpotent simple derivation of the ring such that if , then for some . The derivation is given explicitly, and it is unique up to the action of the group of ring automorphisms of . Let be the set of all such derivations. Then . The proof is based on existence and uniqueness of an iterative -descent (for each ), i.e., a sequence in such that , and for all . For each , and .

     

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.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 604–610, May, 1990.     

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.     

On free group algebras in division rings with uncountable center

Let be a division algebra with uncountable center . If contains a noncommutative free -algebra, then also contains the -group algebra of the free group of rank 2.

     

Group rings of Noetherian groups

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

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.     

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.     

Embeddings of quotient division algebras of rings of differential operators

Let k be an algebraically closed field of characteristic zero, let X and Y be smooth irreducible algebraic curves over k, and let D(X) and D(Y) denote respectively the quotient division rings of the ring of differential operators of X and Y. We show that if there is a k-algebra embedding of D(X) into D(Y), then the genus of X must be less than or equal to the genus of Y, answering a question of the first-named author and Smoktunowicz.     

Noetherian PI Hopf algebras are Gorenstein

We prove that every noetherian affine PI Hopf algebra has finite injective dimension, which answers a question of Brown (1998).

     

Injective resolutions of Noetherian rings and cogenerators

We give new construction of injective resolutions of complexes and bimodules. Applying this construction to an injective resolution of a Noetherian ring, we construct a -embedding cogenerator for the category of modules of projective dimension . Moreover, for a Noetherian projective -algebra , we show that satisfies the Auslander condition if and only if the flat dimension of every -module is equal to or larger than the one of the injective hull .