Based on the work of D. Happel, I. Reiten and S. Smalø on quasitilted artin algebras, the first two authors recently introduced the notion of quasitilted rings. Various authors have presented examples of quasitilted artin algebras that are not tilted. Here we present a class of right quasitilted rings that not right tilted, and we show that they satisfy a condition that would force a quasitilted artin algebra to be tilted.

We prove here an Eilenberg type theorem: the so-called conjunctive varieties of rational languages correspond to the pseudovarieties of finite semilattice-ordered monoids. Taking complements of members of a conjunctive variety of languages we get a so-called disjunctive variety. We present here a non-trivial example of such a variety together with an equational characterization of the corresponding pseudovariety.

In continuation of our article in Colloq. Math. 116.1, we give a complete description of the symmetric algebras of strictly canonical type by quivers and relations, using Brauer quivers.

Assume that K is an arbitrary field. Let (I, ⪯) be a two-peak poset of finite prinjective type and let KI be the incidence algebra of I. We study sincere posets I and sincere prinjective modules over KI. The complete set of all sincere two-peak posets of finite prinjective type is given in Theorem 3.1. Moreover, for each such poset I, a complete set of representatives of isomorphism classes of sincere indecomposable prinjective modules over KI is presented in Tables 8.1.

Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let ${\Gamma }_I\left(R\right)$ be a graph with the set of vertices $S_I\left(R\right)=\{x\in R\setminus I:x+y\in I$ for some $y\in R\setminus I\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in I$. We look at the diameter and girth of this graph. Also we discuss when ${\Gamma }_I\left(R\right)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

Let $m>1,s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p=p\left(x\right)\ge 0,q=q\left(x\right)\ge 0,n=n\left(x\right)\ge 0,r=r\left(x\right)\ge 0$ such that either $x^p[x^n,y]x^q=x^r[x,y^m]y^s$ or $x^p[x^n,y]x^q=y^s[x,y^m]x^r$ for all $y\in R$. In the present paper it is shown that $R$ is commutative if it satisfies the property $Q\left(m\right)$ (i.e. for all $x,y\in R,m[x,y]=0$ implies $[x,y]=0$).

A complete list of positive Tits-sincere one-peak posets is provided by applying combinatorial algorithms and computer calculations using Maple and Python. The problem whether any square integer matrix $A\in ₙ\left(\mathbb{Z}\right)$ is ℤ-congruent to its transpose $A^{tr}$ is also discussed. An affirmative answer is given for the incidence matrices $C_I$ and the Tits matrices $C{̂}_I$ of positive one-peak posets I.