We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the $ω$-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of $n \times n$ upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.
The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then anoncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that (i) in dimension two, all noncommutative quasi-resolutions of a given non-commutative algebra are Morita equivalent, and (ii) in dimension three, all noncommutative quasi-resolutions of a give…
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 clas…
Let $K$ be an infinite field and $K< X> =K< X_1,...,X_n>$ the free associative algebra generated by $X=\{X_1,...,X_n\}$ over $K$. It is proved that if $I$ is a two-sided ideal of $K< X>$ such that the $K$-algebra $A=K< X> /I$ is almost commutative in the sense of [3], namely, with respect to its standard $\mathbb{N}$-filtration $FA$, the associated $\mathbb{N}$-graded algebra $G(A)$ is commutative, then $I$ is generated by a finite Gröbner basis. Therefor, every quotient algebra of the enveloping algebra $U(\mathbf{g})$ of a finite dimensional $K$-Lie algebra $\mathbf{g}$ is, as a noncommutati…
We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization.
Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factor…
A domain $R$ is said to have the finite factorization property if every nonzero non-unit element of $R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let $k$ be an algebraically closed field and let $A$ be a $k$-algebra. We show that if $A$ has an associated graded ring that is a domain with the property that the dimension of each homogeneous component is finite then $A$ is a finite factorization domain. As a corollary, we show that many classes of algebras have the finite factorization property, including Weyl …
We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear representations. To be more specific, a correspondence between factorizations of an element and upper right blocks of zeros in the system matrix (of its representation) is established. The problem is then reduced to solving a system of polynomial equations (with at most quadratic terms) with commuting unknowns to compute appropriate transformation matrices (if p…
This is a survey of recently published results. We introduce and study a wide class algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many well-known graphs such algebras are Koszul and compute their Hilbert series.
Several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded $k$-algebra; these notions are justified by examples from noncommutative invariant theory.
Recently is has been proved that if $σ\in GL_n(R)$ where $R$ is an commutative ring and $n\geq 3$, then each of the elementary transvections $t_{kl}(σ_{ij})~(i\neq j,k\neq l)$ is a product of eight $E_n(R)$-conjugates of $σ$ and $σ^{-1}$. In this article we show that similar results hold true if $R$ is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.
This is an extended version of a talk given at the conference "Algebra and Topology in Interaction" on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995 by I. Gelfand and the author.
We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially ``sewn together'' to form Spec R. In particular, we construct a bimodule-determined functor Mod Z -> Mod R, for a suitable commutative noetherian ring Z, from which there follows a finite-to-one, continous surjection Spec Z -> Spec R. Algebras satisfying the given axiomatic framework include PI algebras finitely generated over fields, noetherian PI algebras, envelo…
The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.
We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers.
The purpose of this survey paper is to bring to a large mathematical audience (containing also non-algebraists) some topics of invariant theory both in the classical commutative and the recent noncommutative case. We have included only several topics from the classical invariant theory -- the finite generating (the Endlichkeitssatz) and the finite presenting (the Basissatz) of the algebra of invariants, the Molien formula for its Hilbert series and the Shephard-Todd-Chevalley theorem for the invariants of a finite group generated by pseudo-reflections. Then we give analogues of these results f…
Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition. In this note we attempt an introductory investigation of totally (also partially) ordered monoids, not assumed commutative (still writing them additively), and taking them as positive, i.e.\ every element is greater than the unit element. That, in the usual commutative cases, allows the ordering to be defined via the algebraic structure, namely, as divisibil…
In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but subsumes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introd…
This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebras to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show th…
For a finite ring $R$, not necessarily commutative, we prove that the category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an application, we prove a very general twisted homology stability for $\text{GL}_n(R)$ with $R$ a finite noncommutative ring.
