### Prufer modules over Leavitt path algebras

Thời gian: 14:00 đến 15:30 ngày 15/05/2018, 14:00 đến 15:30 ngày 22/05/2018, 09:30 đến 11:00 ngày 24/05/2018,

Địa điểm: C2-714, VIASM

Báo cáo viên: Gene Abrams

Tóm tắt:

For a prime $p$ in $\mathbb{Z}$, the well-known and well-studied abelian  group $G = \mathbb{Z}({p^\infty})$ can be constructed.   Effectively, $G$ is the direct union of the groups $\mathbb{Z}/p\mathbb{Z} \subseteq \mathbb{Z}/p^2\mathbb{Z}\subseteq \cdots$.  \ $G$ is called the  {\it Pr\"{u}fer}  $p$-group.   $G$ is divisible, and therefore injective as a $\mathbb{Z}$-module.    Moreover, the only proper subgroups of $G$ are precisely these $\mathbb{Z}/p^i\mathbb{Z}$.

Now let $E$ be a finite graph and $c$ a cycle in $E$.  Starting with the Chen simple $L_K(E)$-module $V_{[c^\infty]}$, we mimic the $\mathbb{Z}({p^\infty})$ construction to produce a direct limit of $L_K(E)$-modules, which we call a {\it Pr\"{u}fer module}, and denote by $U_{E,c-1}$.   In this talk I'll present some properties of these Pr\"{u}fer modules.  Specifically, we give necessary and sufficient conditions on $c$ so that $U_{E,c-1}$ is injective.   We also describe the endomorphism ring of $U_{E,c-1}$. (This is joint work with F. Mantese and A. Tonolo)