This talk will provide a snapshot of contemporary commutative algebra. In classical commutative algebra and algebraic number theory, the Dedekind domains are the most important class of rings. Modern commutative algebra studies numerous generalizations of the Dedekind domains in attempts to generalize results of algebraic number theory. This talk will introduce a few important generalizations o...

Using divided differences associated with the orthogonal groups, we investigate the structure of the polynomial rings over the rings of invariants of the corresponding Weyl groups. We study in more detail the action of orthogonal divided differences on some distinguished symmetric polynomials (P̃ polynomials) and relate it to vertex operators. Relevant families of orthogonal Schubert polynomials...

A skew generalized power series ring R[[S, ω]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are...

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X) ∈ F [X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) ∈ F [X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.

In this paper we prove the universal property of skew PBW extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew PBW extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Art...

We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with.

Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k[x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q → k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k(x, y). Although the value groups correspondi...

Let R = k[x1, . . . , xd] be the ring of polynomials in a finite number of variables over a field k and let DR|k be the corresponding ring of k-linear differential operators. The theory of DR|k-modules has been successfully applied in Commutative Algebra in order to study local cohomology modules due to the fact that, despite not being finitely generated as R-modules, they are so when considere...

Pseudomonomials and ideals generated by pseudomonomials (pseudomonomial ideals) are a central object of study in the theory of neural rings and neural codes. In the setting of a polynomial ring, we define the polarization operation ρ sending pseudomonomials to squarefree monomials and a further polarization operation P sending pseudomonomial ideals to squarefree monomial ideals. We show for a p...