##### Department of Mathematics,

University of California San Diego

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### Algebra Colloquium

## Vladimir Kirichenko

#### Kiev University, Ukraine

## Tiled orders and Frobenius rings

##### Abstract:

By a tiled order we mean a right Noetherian prime semiperfect and semidistributive ring with the nonzero Jacobson radical. For example, a serial tiled order $A$ is a Noetherian (but non-Artinian) serial indecomposable ring. A ring $A$ is decomposable if $A = A_1 \times A_2$, otherwise $A$ is indecomposable. Every serial tiled order $A$ is hereditary and Gorenstein, i.e., $inj.dim_A A_a = inj.dim_A {_A} A = 1$. Let $R(A)$ be the Jacobson radical of a tiled order $A$. For any tiled order $A$ there exists a countable set of two sided ideals $I_1 \supset I_2 \supset \dots$, where $R^2 (A) \supset I_1, I_{k+1} \neq I_k$ and all quotient rings $A/I_k$ are Frobenius. For any permutation $\sigma \in S_n$ there exists a Frobenius ring $B$ with the Nakayama permutaion $\sigma$. We consider the exponent matrices of tiled orders, in particular, Gorenstein matrices. We discuss the relations between exponent matrices and quivers of tiled orders, cyclic Gorenstein orders and doubly stochastic matrices, Gorentstein matrices and tiled orders of injective dimension one.

Host: Efim Zelmanov

### January 29, 2007

### 2:00 PM

### AP&M 7218

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