##### Department of Mathematics,

University of California San Diego

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### Algebra Colloquium

## Vladimir Kirichenko

#### Kiev State Univ., Ukraine

## Quivers of associative rings

##### Abstract:

All rings are associative with $1\not = 0$. A ring $A$ is decomposable if $A=A_{1}\times A_{2}$, otherwise $A$ is indecomposable. We consider three types quivers of rings: Gabriel quiver, prime quiver and Pierce quiver. Gabriel quiver and Pierce quiver are defined for semiperfect rings. Let $A$ be an associative ring with the prime radical $Pr(A)$. The factorring $\bar{A} = A/Pr(A)$ is called the diagonal of $A$. We say that a ring $A$ is a $FD$-ring if $\bar{A}$ is a finite direct product of indecomposable rings. We define the prime quiver of $FD$-ring with $T$-nilpotent prime radical. We discuss the properties of rings and its quivers, for example, a right Noetherian semiperfect ring is semisimple Artinian if and only if its Gabriel quiver is a disconnected union of vertices (without arrows).

Host: Efim Zelmanov

### March 17, 2009

### 3:00 PM

### AP&M 6218

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