Generalized Koszul algebra and Koszul duality

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Let [Formula: see text] be a subset of a commutative graded ring [Formula: see text]. The Cayley graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The Cayley sum graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the set of homogeneous elements and [Formula: see text] be the set of zero-divisors of [Formula: see text]. In this paper, we study [Formula: see text] (total graph) and [Formula: see text]. In particular, if [Formula: see text] is an Artinian graded ring, we show that [Formula: see text] is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of [Formula: see text] for some finite graded ring [Formula: see text].

Let $R$ be a Cohen-Macaulay ring and $M$ a maximal Cohen-Macaulay $R$-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and van den Bergh we study the question of when the endomorphism ring of $M$ has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced $1$-dimensional hypersurfaces.

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Recently, the study of probabilities in ring theory has shown a significant increase in the field of algebra. Many interesting algebraic structures were modeled to find their probabilities in certain finite rings. In this paper, we introduce a new type of probability in finite rings, namely the squared-zero product probability. The aim is to study the square property and the zero product property of the ring. The focus of this study is the ring of matrices of dimension two over integers modulo prime p. To obtain the probability, the order of the square-annihilator of the ring is determined. The results found show that the squared-zero product probability of the ring is dependent on the value of p.

A primary problem in the theory of quadratic forms over a field F of characteristic different from two is to prove that the rings H'F and GWF are isomorphic.Here H'F = H'(Ga\(Fq/F),Z/2Z)),where Fq is the quadratic closure of F , and GWF is the graded Witt ring associated to the fundamental ideal of even dimensional forms in the Witt ring WF of F .In this paper, we assume we are given a field extension K of F such that WK is 'close' to WF or H'K is 'close' to H'F .A method is developed to obtain information about these graded rings over F and its 2-extensions from information about the corresponding graded ring of K.This relative theory extends and includes the previously developed absolute case where K = Fq .Applications are also given to show that H'F and GWF are isomorphic for a collection of fields arising naturally from the theory of abstract Witt rings.

Let R be a commutative ring with a nonzero unity element. In this article, we introduce and examine new classes of ideals in Z2-graded rings, expanding upon the earlier concept of N-prime ideals. Using the function N : R → R0, defined by N(x) = x0 2 – x1 2 for x = x0 + x1 ∈ R, we define and analyze two new types of ideals, 2-N-prime and weakly 2-N-prime ideals. A proper ideal I is called 2-N-prime if xy ∈ I implies that N(x2) ∈ I or N(y2) ∈ I. Similarly, I is weakly 2-N-prime if 0 ≠ xy ∈ I implies that N(x2) ∈ I or N(y2) ∈ I. We explore the fundamental properties of these ideals, highlighting their connections to established structures in graded ring theory. Through examples and counter examples, we illustrate the differences between 2-N-prime and weakly 2-N-prime ideals, underscoring their significance in the theory of graded rings. These results not only deepensour understanding of ideal theory in Z2-graded rings but also open new avenues for futures research.

Fully bounded grothendieck categories part II: Graded modules

In this note we first study regular Z \mathbb {Z} -graded local rings. We characterize commutative noetherian regular Z \mathbb {Z} -graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated singularity for a commutative Z \mathbb {Z} -graded semilocal algebra in terms of the global dimension of its associated noncommutative projective scheme. As a corollary, we obtain that a commutative affine N \mathbb {N} -graded algebra generated in degree 1 1 is a graded isolated singularity if and only if its associated noncommutative projective scheme is smooth; if and only if the category of coherent sheaves on its projective scheme has finite global dimension, which are known in literature.

A Galois theory of commutative rings

Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative N-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative N-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.

We give a negative answer to a question posed by A. V. Jategaonkar: is it not true that an arbitrary primary principal left ideal ring is a factor of a prime principal left ideal ring? We give a counter example in the class of finite complete primary principal ideal rings, the so-called Galois-Eisenstein-Ore rings.

We prove that ifR is a commutative ring such that each localization at a prime ideal has finite weak global dimension then everyR-module has a flat envelope if and only ifR is coherent and has weak global dimension less than or equal to two.

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.