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On the classification of rings of “small ” rank

By Manjul Bhargava


The purpose of these notes is to study rings of rank n: Definition 1 A ring of rank n is a commutative, associative ring with identity that is free of rank n as a Z-module. That is, a ring of rank n is a ring (commutative, associative, with unit) whose underlying additive group is isomorphic to Z n. The prototypical examples of rings of rank n are, of course, orders in degree n number fields. The class of all such examples consists precisely of those rings of rank n that are integral domains. However, there are many interesting examples of rings of rank n that are not integral domains. For example, there are degenerate rings such as Z[x]/(x n) or Z[x1,...,xn−1]/(x1,...,xn−1) 2. One may also construct rings of rank n by taking (any rank n subring of) a direct sum of k rings having ranks n1,...,nk respectively, where n1 + · · · + nk = n. For instance, Z ⊕n is a nice example of a ring of rank n. More generally, we may consider rings of rank n over any base ring T: a ring of rank n over T is any ring that is locally free of rank n as a T-module. Concerning terminology, we refer to rings of rank 2, 3, 4, 5, or 6 as quadratic, cubic, quartic, quintic, or sextic rings respectively. In these notes, we wish to classify rings of small rank n, where by “small ” we mean “at most 5”. 1 We begin with the simplest possible case, namely n = 1.

Year: 2009
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