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 $P_n$ in $n$ variables by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of $P_n$. 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 $2n$; but the weak and the global dimensions of \mS_n are $n$. 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 $n$-filters. These are related to the finite Boolean lattice with $n$ atoms in the same way that filters are related to the two-element Boolean lattice: $n$-filters are precisely the intersections of prime $n$-filters and prime $n$-filters are precisely the homomorphic preimages of the prime $n$-filter of non-zero elements of the finite Boolean lattice with $n$ atoms. Moreover, $n$-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 $\mathcal{V}$ allows us to define the prime congruences of any algebra $A\in \mathcal{V}$ and the prime spectrum $Spec(A)$ of $A$. 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 $A\in \mathcal{V}$ is a bounded distributive algebra $L(A)$, whose prime spectrum (endowed with the Stone topology) is homeomorphic to $Spec(A)$ (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 $A$ in a semidegenerate congruence-modular variety $\mathcal{V}$, satisfying the hypothesis $(H)$: the set $K(A)$ of compact congruences of $A$ 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 $\mathcal{V}$ 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 $K$-algebra $R$ are parameterized by the elements of cotangent module $T^1(R)$ of $R$. In this article we focus on deformations of toric rings, and give an explicit description of $T^1(R)$ in the case that $R$ 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 $m-k=k=3$, $G_{k,m-k}$ is not rigid while for $m-k\geq k\geq 4$, $G_{k,m-k}$ is rigid. Here $G_{k,m-k}$ is the complete bipartite graph $K_{m-k,k}$ 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 $E$ is a directed graph, $K$ is a field, and $I$ is a graded ideal of the Leavitt path algebra $L_K(E),$ $I$ is completely determined by an admissible pair $(H,S)$ of two sets of vertices of $E$. The ideal $I=I(H,S)$ is graded isomorphic to the Leavitt path algebra of the {\em porcupine graph} of $(H,S)$ and the quotient $L_K(E)/I$ is graded isomorphic to the Leavitt path algebra of the {\em quotient graph} of $(H,S).$ We present a construction which generalizes both constructions and enables one to consider quotients of graded ideals: if $(H,S)$ and $(G,T)$ are admissible pairs such that $I(H,S)\subseteq I(G,T)$, we define the {\em porcupine-quotient graph} $(G,T)/(H,S)$ such that its Leavitt path algebra is graded isomorphic to the quotient $I(G,T)/I(H,S).$ Using the porcupine-quotient construction, the existence of a graded composition series of $L_K(E)$ is equivalent to the existence of a finite chain of admissible pairs of $E,$ 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 $E$ is equivalent to the existence of a composition series of the graph monoid $M_E$ as well as a composition series of the talented monoid $M_E^\Gamma.$ An ideal of $M_E^\Gamma$ is minimal exactly when it is generated by the element of $M_E^\Gamma$ corresponding to a terminal vertex.Comment: Example 2.2 added and some minor changes made to address the comments of Publicacions Matem\`atiques referee