Rational Modules for Corings

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.

Resolutions of determinantal ideals: The submaximal minors

The various canonical subcategories of the category HopfR of Hopf algebras over a commutative ring R, like those of (co)commutative Hopf algebras or Hopf algebras whose antipode is bijective or of order 2, are shown to be locally presentable categories and reflective and coreflective in their respective supercategories. The reflectivity results provided only hold for commutative von Neumann regular rings, while most of the coreflectivity results are valid over any ring. As a consequence one gets existence of free commutative Hopf algebras over coalgebras and cofree cocommutative Hopf algebras over algebras.

This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of $\\mathbb{Z}$ -modules, where the radical of a $\\mathbb{Z}$ -module $M$ is defined as the intersection of $pM=\\{px:x\\in M\\}$ with $p$ taken from all primes. It shows that $\\mathrm{ACA}_{0}$ is equivalent to the existence of radicals of $\\mathbb{Z}$ -modules over $\\mathrm{RCA}_{0}$ . We then study socles of modules over commutative rings with identity. The socle of an $R$ -module $M$ is the largest semisimple submodule of $M$ . We show that the existence of socles of modules over a commutative ring with identity is equivalent to $\\mathrm{ACA}_{0}$ over $\\mathrm{RCA}_{0}$ . Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

We give a characterization theorem for the E(A)-normalized subgroups of G2 (A), where A is any commutative ring. This is the last of the simple Chevalley-Demazure group-schemes for which such a theorem is lacking. INTRODUCTION In this paper we prove a theorem (3.6) which characterizes the E(A)-normalized subgroups of G2(A) for an arbitrary commutative ring A. The theorem brings to completion the larger project of finding characterization theorems for E(A)normalized subgroups of simple Chevalley-Demazure group-schemes G(A) of rank > 2 over arbitrary commutative rings. The bulk of t;he project was done by Vaserstein in his essential paper [16]. There he proved the standard characterization theorem, given by a ladder corresponding to an ideal, for all commutative rings for all G of rank > 2 except those of types B2= C2 and G2. Vaserstein also suggested in that paper that previous work by Abe [1] and Abe-Suzuki [3] could be improved to yield more. This was accomplished by Abe in [2], wherein he showed that E(A)normalized subgroups of G(A) could be characterized by ladders determined by pairs of ideals, with the exception that for G of type B2 = C2 or G2 rings A having residue class fields of two elements must be excluded. Our paper [8] subsequently succeeded in removing this restriction for G of type C2 (and so also for G of type B2) by recognizing that the presence of residue fields of two elements required lowering the bottom of the ladder. This left only the case G of type G2 to be finished; the present paper does so by applying the lower bottom rung idea of [8] (see also [7]) to this case. We now give a more explicit description of our results. Let A be an arbitrary commutative ring and denote by G2(A) the group of type G2 defined over A. The main theorem (3.6) is that a subgroup N of G2(A) is E(A)-normalized if and only if C(J, J') C N C G(J, J') for a pair of ideals J, J' satisfying the condition J3 C J' C J. Here, J3 denotes the ideal of A generated by all elements of the form x3 and 3x for all x in J, so that (J, J') is an admissible pair in the sense of Abe. The bracketing groups C(J, J') and G(J, J') are defined in Section 1 below. In essence, C(J, J') is the mixed commutator group [E(A), E(J, J')], where E(J, J') is the E(A)-normalized subgroup generated by all elements x,(j) with a long and Received by the editors April 1, 1997 and, in revised form, May 22, 1997. 1991 Mathematics Sutbject Classification. Primary 20H05; Secondary 20G35. Research partially supported by NSA grant MDA 904-94-H-2008 and NSF grant DMS-9622899. (?)1999 American Mathematical Society

Block-form control of linear time-invariant discrete-time systems defined over commutative rings

The computational complexity of module socles

The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring

ABSTRACTThe set Fil RR of all right topologizing filters on a fixed, but arbitrary ring R is both a complete lattice under inclusion and a monoid with respect to an order compatible, but in general noncommutative, binary operation. It is known that the order dual of Fil RR is always left residuated, meaning, for each pair 𝔉,𝔊∈Fil RR, there exists a smallest ℌ∈Fil RR such that ℌ:𝔊⊇𝔉, but is not, in general, right residuated (there exists a smallest ℌ such that 𝔊:ℌ⊇𝔉). This paper investigates rings R for which is both left and right residuated. It is shown that every right fully bounded Noetherian ring possesses this property and that a commutative ring will too if and only if the binary operation : on Fil RR is commutative. This has the consequence that if R is a valuation domain, then will be left and right residuated precisely when R is rank one discrete.

In this article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt path algebra LR(E) is a direct sum of minimal basic ideals and that for a commutative ring R and a graph E satisfying Condition (L), the Leavitt path algebra LR(E) has no non-zero nilpotent basic ideals. Uniqueness theorems for Leavitt path algebras over commutative unital rings are also discussed.

Affine Chevalley algebras

This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

Assignable polynomials to linear systems over von Neumann regular rings

This is a survey of results describing the parabolic subgroups of Chevalley groups over rings of various types. For Chevalley groups of classical types over an arbitrary commutative ring, a description of parabolic subgroups is obtained which may be considered definitive. Some errors in the results of V. M. Levchuk in Mat. Zametki, Volume 31, No. 4 (1982) are pointed out.