Factorizations of Elements in Noncommutative Rings: A Survey

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.Comment: 50 pages, comments welcom

Leavitt path algebras: the first decade

The algebraic structures known as {\it Leavitt path algebras} were initially developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author and Aranda Pino. During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in C$^*$-algebras, group theorists, and symbolic dynamicists as well. The goal of this article is threefold: to introduce the notion of Leavitt path algebras to the general mathematical community; to present some of the important results in the subject; and to describe some of the field's currently unresolved questions.Comment: 53 pages. To appear, Bulletin of Mathematical Sciences. (page numbering in arXiv version will differ from page numbering in BMS published version; numbering of Theorems, etc ... will be the same in both versions

On K_0 of locally finte categories

We calculate the Grothendieck group $K_0(\cal A)$, where $\cal A$ is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category $\mathsf{SW}$ of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category $\cal A$ and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd's theorem describing the group $K_0(\mathsf{SW})$.Comment: 21 page

Notes on Simple Modules over Leavitt Path Algebras

Given an arbitrary graph E and any field K, a new class of simple left modules over the Leavitt path algebra L of the graph E over K is constructed by using vertices that emit infinitely many edges. The corresponding annihilating primitive ideals are described and is used to show that these new class of simple L-modules are different from(that is non-isomorphic to) any of the previously known simple modules. Using a Boolean subring of idempotents induced by paths in E, bounds for the cardinality of the set of distinct isomorphism classes of simple L-modules are given. We also append other information about the Leavitt path algebra L(E) of a finite graph E over which every simple left module is finitely presented.Comment: 17 page

Leavitt path algebras: Graded direct-finiteness and graded Σ\Sigma-injective simple modules

In this paper, we give a complete characterization of Leavitt path algebras which are graded $\Sigma$-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $\Sigma$-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical characterization of such a graded $\Sigma$-$V$ ring $L$% . When the graph $E$ is finite, we show that $L$ is a graded $\Sigma$-$V$ ring $\Longleftrightarrow L$ is graded directly-finite $\Longleftrightarrow L$ has bounded index of nilpotence $\Longleftrightarrow$ $L$ is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph $E$ is infinite. Following this, we also characterize Leavitt path algebras $L$ which are non-graded $\Sigma$-$V$ rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra $L$ is a graded $\Sigma$-$V$ ring, then $L$ is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Va\v{s} \cite{V} on directly-finite Leavitt path algebras.Comment: 21 page

Quivers of monoids with basic algebras

We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as $\mathbf{DO}$) to representation theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of $\mathscr R$-trivial monoids, we also provide a semigroup theoretic description of the projective indecomposables and compute the Cartan matrix.Comment: Minor corrections and improvements to exposition were made. Some theorem statements were simplified. Also we made a language change. Several of our results are more naturally expressed using the language of Karoubi envelopes and irreducible morphisms. There are no substantial changes in actual result

An exact category approach to Hecke endomorphism algebras

Let $G$ be a finite group of Lie type. In studying the cross-characteristic representation theory of $G$, the (specialized) Hecke algebra H=\End_G(\ind_B^G1_B) has played a important role. In particular, when $G=GL_n(\mathbb F_q)$ is a finite general linear group, this approach led to the Dipper-James theory of $q$-Schur algebras $A$. These algebras can be constructed over \sZ:=\mathbb Z[t,t^{-1}] as the $q$-analog (with $q=t^2$) of an endomorphism algebra larger than $H$, involving parabolic subgroups. The algebra $A$ is quasi-hereditary over \sZ. An analogous algebra, still denoted $A$, can always be constructed in other types. However, these algebras have so far been less useful than in the $GL_n$ case, in part because they are not generally quasi-hereditary. Several years ago, reformulating a 1998 conjecture, the authors proposed (for all types) the existence of a \sZ-algebra $A^+$ having a stratified derived module category, with strata constructed via Kazhdan-Lusztig cell theory. The algebra $A$ is recovered as $A=eA^+e$ for an idempotent $e\in A^+$. A main goal of this monograph is to prove this conjecture completely. The proof involves several new homological techniques using exact categories. Following the proof, we show that $A^+$ does become quasi-hereditary after the inversion of the bad primes. Some first applications of the result -- e.g., to decomposition matrices -- are presented, together with several open problems.Comment: 144 page