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The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…

We study the representation theory of three towers of algebras which are related to the symmetric groups and their Hecke algebras. The first one is constructed as the algebras generated simultaneously by the elementary transpositions and…

The paper is devoted to the investigation of finite dimensional commutative nilpotent (associative) algebras N over an arbitrary base field of characteristic zero. Due to the lack of a general structure theory for algebras of this type (as…

Let $R=\oplus_{\Gamma\in\Gamma}R_{\gamma}$ be a $\Gamma$-graded $K$-algebra over a field $K$, where $\Gamma$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $\Gamma$-grading filtration $FR$ of $R$ and the…

The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous $\Bbbk$-subalgebra of a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. Clearly, any affine…

We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a…

We introduce FI-algebras over a commutative ring $K$ and the category of FI-modules over an FI-algebra. Such a module may be considered as a family of invariant modules over compatible varying $K$-algebras. FI-modules over $K$ correspond to…

We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a…

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive…

In this paper, we define on one hand, the notions of characteristics as well as central characteristics ideals of a given Leibniz algebra g and provide a necessary condition under which for two given subalgebras J and K of g such that, J IS…

Let T be a complete local (Noetherian) ring and let A be a local subring of T such that the completion of A with respect to its maximal ideal is T. We investigate the possible structures of the partially ordered set Spec(A). Specifically,…

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard…

For every partially ordered sets I, having a finite cofinal subset, and every field K we build a unital, locally matricial and hence unit-regular K-algebra B(I) such that the lattice of all its ideals is order isomorphic to the lattice of…

A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite…

The face monoid described in [M1] acts on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid whose unit group is a Kac-Moody group. We found in…

We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…

This paper is concerned with the prime spectrum of a tensor product of algebras over a field. It seeks necessary and sufficient conditions for such a tensor product to have the S-property, strong S-property, and catenarity. Its main results…

Bartholdi and Smoktunowicz constructed finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras…

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes we conclude that every such algebra is normal and Cohen-Macaulay, and give an interpretation of its…