关于单oid格子半赫瑞迪环的特征

The graded structure of algebraic Cuntz-Pimsner rings

The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We classify strongly, epsilon-strongly and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. As an application, we characterize noetherian and artinian fractional skew monoid rings by a single corner automorphism.

Action of Pontryagin dual of semilattices grading algebras

Let A be a unitary algebra and G be a finite abelian group. Then a G-graded algebra is merely a G-algebra and viceversa because of the fact that G and its group of characters G* are isomorphic. This fact is no longer true if we substitute G with infinite or non-abelian groups. In this paper we try to obtain similar results for a special class of abelian monoids, i.e, the bounded semmilattices. Moreover, if S is such a monoid, we are going to investigate the role of S and its Pontryagin dual S* over the algebra A, in the case A is S-graded.

Köthe's Problem, Kurosch-Levitzki Problem and Graded Rings

Let $\mathfrak{R}$ be an associative ring graded by left cancellative monoid $\mathsf{S}$, and $e$ the neutral element of $\mathsf{S}$. We study the following problem: if $\mathfrak{R}_e$ is nil, then is $\mathfrak{R}$ nil/nilpotent? We have proved that if $\mathfrak{R}_e$ is nil (of bounded index) and $\mathsf{f}$- commutative, then $\mathfrak{R}$ is nil (of bounded index). Later, we have shown that $\mathfrak{R}_e$ being nilpotent implies $\mathfrak{R}$ is nilpotent. Consequently, we have exhibited a generalization of Dubnov-Ivanov-Nagata-Higman Theorem for the graded algebras case. Furtherm…

分级环中零因子、单位和幂等元的齐性

Homogeneity of zero-divisors, units and idempotents in a graded ring

本文受Kaplansky猜想启发，证明了分级环（尤其是monoid-rings）的若干重要结果，将群环 $K[G]$ 的Kaplansky零因子猜想推广至无挠群 $G$ 的 $G$-分级环情形。

Skew polynomial rings, Groebner bases and the letterplace embedding of the free associative algebra

In this paper we introduce an algebra embedding $ι:K< X >\to S$ from the free associative algebra $K< X >$ generated by a finite or countable set $X$ into the skew monoid ring $S = P * Σ$ defined by the commutative polynomial ring $P = K[X\times N^*]$ and by the monoid $Σ= < σ>$ generated by a suitable endomorphism $σ:P\to P$. If $P = K[X]$ is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra $S = \bigoplus_i S_i$ with $S_i = P σ^i$ and $σ:P \to P$ an abstract endomorphism satisfyi…

The Hunting of the Hopf Ring

We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E^*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E^*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.

Fractional skew monoid rings

Given an action of a monoid $T$ on a ring $A$ by ring endomorphisms, and an Ore subset $S$ of $T$, a general construction of a fractional skew monoid ring $S^{\rm op} * A * T$ is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case $S$ is a subsemigroup of a group $G$ such that $G=S^{-1}S$, we obtain a $G$-graded ring $S^{\rm op} * A * S$ with the property that, for each $s\in S$, the $s$-component contains a left invertible element and the $s^{-1}$-component contains a right invertible element. In the most basic case, where $G$ is the additive grou…

Algebras and their Associated Monomial Algebras

Let $R=\oplus_{Γ\inΓ}R_γ$ be a $Γ$-graded $K$-algebra over a field $K$, where $Γ$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $Γ$-grading filtration $FR$ of $R$ and the $Γ$-filtration $FA$ induced by $FR$ for the quotient $K$-algebra $A=R/I$, we show that there is a $Γ$-graded $K$-algebra isomorphism $G(A)\cong \bar{A}=R/<\hbox{\bf HT} (I)>$, where $G(A)$ is the associated $Γ$-graded $K$-algebra of $A$ defined by $FA$, and $<\hbox{\bf HT}(I)>$ is the $Γ$-graded ideal of $R$ generated by the set of head terms of $I$. In the case that $Γ$ is an ordered monoid …

Nonstable $K$-theory for graph algebras

We compute the monoid $V(L_K(E))$ of isomorphism classes of finitely generated projective modules over certain graph algebras $L_K(E)$, and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of $L_K(E)$ and the lattice of order-ideals of $V(L_K(E))$. When $K$ is the field $\mathbb C$ of complex numbers, the algebra $L_{\mathbb C}(E)$ is a dense subalgebra of the graph $C^*$-algebra $C^*(E)$, and we show that the inclusion map induces an isomorphism between the correspondin…

群分次环中的中心幂等元

Central idempotents in group-graded rings

本文研究群分次环中非零中心幂等元的支撑群有限性条件，推广了群环和交叉积中的经典结果，并应用于半群分次环、Leavitt路环等结构。