# Seminario MALGA Padova Verona - Moduli Algebre Anelli: When is the stable category abelian? + Integer-valued Polynomials on Noncommutative Rings and Algebra

## Giovedì 21 Maggio 2015, ore 15:30 - Aula 2AB40 - Alex Martsinkovsky + Nicholas Werner

ARGOMENTI: Seminari

Giovedì 21 Maggio 2015 in Aula 2AB40 si terrà un incontro del Seminario Padova - Verona MALGA Moduli Algebre Anelli

ore 15:30
Alex Martsinkovsky (Northeastern University - Boston)
"When is the stable category abelian?"
Abstract.
We will provide a complete answer to the above question for left hereditary rings. More importantly, we observe tight and rather surprising connections between the properties of the ring and the properties of its (projectively) stable module category.
This is joint work with Dali Zangurashvili.

ore 16:45
Nicholas Werner (Ohio State University - Newark)
"Integer-valued Polynomials on Noncommutative Rings and Algebras"
Abstract
Given a commutative integral domain $D$ with field of fractions $k$, the classical ring of integer-valued polynomials over $D$ is $\mathrm{Int}(D) := \{f (x) \in k[x] \mid f (D) \subseteq D\}$. We will discuss a generalization of this construction to sets of polynomials that act on $D$-algebras. When $A$ is a torsion-free $D$-algebra that is finitely generated as a $D$-module, we define $\textrm{Int}(A) := \{f (x) \in B[x] \mid f (A) \subseteq A\}$, where $B$ is the extension of $A$ to a $k$-algebra. The set $\mathrm{Int}(A)$ consists of polynomials with non-commuting coefficients and is always a left $A$-module. In certain cases (such as when $A$ is a matrix algebra or - more generally - is generated by units) $\mathrm{Int}(A)$ is actually a ring, although it is not known whether this is true in general. We will discuss the current known theorems regarding $\mathrm{Int}(A)$ and other associated objects; present interesting connections that arise involving commutative rings, noncommutative rings, and finite rings; and mention open problems and possibilities for future research.