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In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element $h$ and obtain families of such simple modules of arbitrary rank. In…

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K represented by central division algebras D' of degree n over K having the same…

There are several remarks on Hilbert series of finitely presented (f. p.) associative algebras over a field and their modules. First, given an integer $D$, the set of Hilbert series of right-sided ideals with generators and relations of…

In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two…

For every variety of algebras and every algebras in these variety we can consider an algebraic geometry. Algebras may be many sorted (not necessarily one sorted) algebras. A set of sorts is fixed for each variety. This theory can be applied…

We call a graded connected algebra $R$ effectively coherent, if for every linear equation over $R$ with homogeneous coefficients of degrees at most $d$, the degrees of generators of its module of solutions are bounded by some function…

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce…

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension.…

The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…

Various concepts associated with quadratic algebras admit natural generalizations when the quadratic algebras are replaced by graded algebras which are finitely generated in degree 1 with homogeneous relations of degree N. Such algebras are…

Let $A$ be a not necessarily commutative monoid with zero such that projective $A$-acts are free. This paper shows that the algebraic K-groups of $A$ can be defined using the +-construction and the Q-construction. It is shown that these two…

The $\mathcal{R}$-height of a semigroup $S$ is the height of the poset of $\mathcal{R}$-classes of $S,$ i.e. the supremum of the lengths of chains of $\mathcal{R}$-classes. Given a semigroup $S$ with finite $\mathcal{R}$-height, we…

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero…

Let A be a connected graded noncommutative monomial algebra. We associate to A a finite graph \Gamma(A) called the CPS graph of A. Finiteness properties of the Yoneda algebra Ext_A(k,k) including Noetherianity, finite GK dimension, and…

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up…

In [14] we introduced a new class of algebras, which we named \textit{quantum generalized Heisenberg algebras} and which depend on a parameter $q$ and two polynomials $f,g$. We have shown that this class includes all generalized Heisenberg…

In this paper, we introduce the class of finitely semi-graded algebras which extends the connected graded algebras finitely generated in degree one. The Koszul behavior of finitely semi-graded algebras is investigated by the distributivity…

Let $f\_1,\ldots, f\_s$ be formal power series (respectively polynomials) in thevariable $x$. We study the semigroup of orders of the formal series inthe algebra $K[[ f1,\ldots, f\_s]] \subseteq K[[ x ]]$ (respectively the semigroup of…

Let $E\supseteq F$ be a field extension and $M$ a graded Lie algebra of maximal class over $E$. We investigate the $F$-subalgebras $L$ of $M$, generated by elements of degree $1$. We provide conditions for $L$ being either ideally…

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at…