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## Huge Rings of low rank with a standard involution and quaternion rings

### John Voight

University of Vermont

###
Thursday, October 1, 2009

007 Kemeny Hall, 4 pm

Tea 3:30 pm, 300 Kemeny Hall

**Abstract: **
A ring of rank n is an associative ring with identity that is free of rank n as a module over the integers $\Bbb Z$ (i.e. is isomorphic to $\Bbb Z^n$ as an abelian group). Bhargava and others have studied and classified commutative rings of rank at most 5. We consider the case where the ring is no longer (necessarily) commutative but possesses a standard involution. We characterize such rings by their degree, discuss the relationship to quaternion rings and ternary quadratic forms (due to Gross and Lucianovic), and provide a generalization to an arbitrary base ring.

This talk will be accessible to graduate students.