On Rings Whose Finite Length Indecomposable Modules Are Completely Determined by Their Composition Factors – Dokuz Eylül University Faculty of Science Department of Mathematics

Victor Blasco Jimenez, Dokuz Eylül University.

Date: 1st of March, 2023, Wednesday, Time: 10.30 – 12.00.

Place:Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: From the Fundamental Theorem of Abelian Groups we can deduce that every finite length indecomposable module over $\mathbb{Z}$ is uniquely determined by its composition factors set, meaning that if we have two indecomposable finite length abelian groups $N^1$ and $N^2$ with the same length and, if for any $0\subseteq N_{1}^j\subseteq ... \subseteq N_{r}^j=N^j$ composition series for $N^j$ , $j=1,2$ , we get $\{N_{i+1}^1/N_{i}^1\}_{i=0}^{r-1}=\{N_{i+1}^2/N_{i}^2\}_{i=0}^{r-1}$ , then we must have $N^1\cong N^2$ . In this series of talks , we will study this property about the finite length indecomposable abelian groups in a more general way. We will start by focusing on the class of commutative rings $R$ which satisfy it, showing that it contains the class of Dedekind Domains. If time permits, we will see that if $R$ is any unital ring (not necessarily commutative) satisfying this property, which we will call “Property $\mathfrak{X}$ “, and $I$ is an ideal of $R$ , then also $R/I$ satisfies it. This work is part of my ongoing master thesis “Some methods of Category Theory in the Representation Theory of Artin Algebras”.