190 research outputs found
The algebra of one-sided inverses of a polynomial algebra
We study in detail the %Shrek algebra \mS_n in the title which is an
algebra obtained from a polynomial algebra in variables by adding
commuting, {\em left} (but not two-sided) inverses of the canonical generators
of . The algebra \mS_n is non-commutative and neither left nor right
Noetherian but the set of its ideals satisfies the a.c.c., and the ideals {\em
commute}. It is proved that the classical Krull dimension of \mS_n is ;
but the weak and the global dimensions of \mS_n are . The prime and
maximal spectra of \mS_n are found, and the simple \mS_n-modules are
classified. It is proved that the algebra \mS_n is central, prime, and {\em
catenary}. The set \mI_n of idempotent ideals of \mS_n is found explicitly.
The set \mI_n is a finite distributive lattice and the number of elements in
the set \mI_n is equal to the {\em Dedekind} number \gd_n.Comment: 33 page
Filter classes of upsets of distributive lattices
Let us say that a class of upward closed sets (upsets) of distributive
lattices is a finitary filter class if it is closed under homomorphic
preimages, intersections, and directed unions. We show that the only finitary
filter classes of upsets of distributive lattices are formed by what we call
-filters. These are related to the finite Boolean lattice with atoms in
the same way that filters are related to the two-element Boolean lattice:
-filters are precisely the intersections of prime -filters and prime
-filters are precisely the homomorphic preimages of the prime -filter of
non-zero elements of the finite Boolean lattice with atoms. Moreover,
-filters on Boolean algebras are the only finitary filter classes of upsets
of Boolean algebras generated by prime upsets.Comment: 24 pages, 2 figure
Reticulation of Quasi-commutative Algebras
The commutator operation in a congruence-modular variety allows
us to define the prime congruences of any algebra and the
prime spectrum of . The first systematic study of this spectrum
can be found in a paper by Agliano, published in Universal Algebra (1993).
The reticulation of an algebra is a bounded distributive
algebra , whose prime spectrum (endowed with the Stone topology) is
homeomorphic to (endowed with the topology defined by Agliano). In a
recent paper, C. Mure\c{s}an and the author defined the reticulation for the
algebras in a semidegenerate congruence-modular variety ,
satisfying the hypothesis : the set of compact congruences of
is closed under commutators. This theory does not cover the Belluce
reticulation for non-commutative rings. In this paper we shall introduce the
quasi-commutative algebras in a semidegenerate congruence-modular variety
as a generalization of the Belluce quasi-commutative rings. We
define and study a notion of reticulation for the quasi-commutative algebras
such that the Belluce reticulation for the quasi-commutative rings can be
obtained as a particular case. We prove a characterization theorem for the
quasi-commutative algebras and some transfer properties by means of the
reticulationComment: arXiv admin note: text overlap with arXiv:2205.0217
On systems of congruences on principal filters of orthomodular implication algebras
summary:Orthomodular implication algebras (with or without compatibility condition) are a natural generalization of Abbott’s implication algebras, an implication reduct of the classical propositional logic. In the paper deductive systems (= congruence kernels) of such algebras are described by means of their restrictions to principal filters having the structure of orthomodular lattices
On categorical equivalence of finite p-rings
We prove that finite categorically equivalent p-rings have isomorphic additive groups (in particular, they have the same cardinality) and that the number of generators is a categorical invariant for finite rings. We also classify rings of size p (3) up to categorical equivalence
Smashing localizations of rings of weak global dimension at most one
none2siWe show for a ring R of weak global dimension at most one that there is a bijection
between the smashing subcategories of its derived category and the equivalence classes of homological
epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly
generated localizing subcategories correspond precisely to flat epimorphisms. We also classify
smashing localizations of the derived category of any valuation domain, and provide an easy criterion
for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a
consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it
holds precisely for those Pruefer domains which are strongly discrete.mixedSilvana Bazzoni; Jan StovicekBazzoni, Silvana; Jan, Stovice
Toric rings, inseparability and rigidity
This article provides the basic algebraic background on infinitesimal
deformations and presents the proof of the well-known fact that the non-trivial
infinitesimal deformations of a -algebra are parameterized by the
elements of cotangent module of . In this article we focus on
deformations of toric rings, and give an explicit description of in
the case that is a toric ring.
In particular, we are interested in unobstructed deformations which preserve
the toric structure. Such deformations we call separations. Toric rings which
do not admit any separation are called inseparable. We apply the theory to the
edge ring of a finite graph. The coordinate ring of a convex polyomino may be
viewed as the edge ring of a special class of bipartite graphs. It is shown
that the coordinate ring of any convex polyomino is inseparable. We introduce
the concept of semi-rigidity, and give a combinatorial description of the
graphs whose edge ring is semi-rigid. The results are applied to show that for
, is not rigid while for , is
rigid. Here is the complete bipartite graph with one
edge removed.Comment: 33 pages, chapter 2 of the Book << Multigraded Algebra and
Applications>> 2018, Springer International Publishing AG, part of Springer
Natur
Porcupine-quotient graphs, the fourth primary color, and graded composition series of Leavitt path algebras
If is a directed graph, is a field, and is a graded ideal of the
Leavitt path algebra is completely determined by an admissible
pair of two sets of vertices of . The ideal is graded
isomorphic to the Leavitt path algebra of the {\em porcupine graph} of
and the quotient is graded isomorphic to the Leavitt path algebra of
the {\em quotient graph} of We present a construction which
generalizes both constructions and enables one to consider quotients of graded
ideals: if and are admissible pairs such that , we define the {\em porcupine-quotient graph} such that
its Leavitt path algebra is graded isomorphic to the quotient
Using the porcupine-quotient construction, the existence of a graded
composition series of is equivalent to the existence of a finite chain
of admissible pairs of starting with the trivial and ending with the
improper pair, such that the quotient of two consecutive pairs is cofinal (a
graph is cofinal exactly when its Leavitt path algebra is graded simple). We
characterize the existence of such a composition series with a set of
conditions which also provides an algorithm for obtaining such a series. The
conditions are presented in terms of four types of vertices which are all
``terminal'' in a certain sense. Three types are often referred to as the three
primary colors and the fourth type is new. As a corollary, a unital Leavitt
path algebra has a graded composition series.
We show that the existence of a composition series of is equivalent to
the existence of a composition series of the graph monoid as well as a
composition series of the talented monoid An ideal of
is minimal exactly when it is generated by the element of
corresponding to a terminal vertex.Comment: Example 2.2 added and some minor changes made to address the comments
of Publicacions Matem\`atiques referee
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