On the classification of rings of “small” rank
نویسنده
چکیده
That is, a ring of rank n is a ring (commutative, associative, with unit) whose underlying additive group is isomorphic to Z. 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) or Z[x1, . . . , xn−1]/(x1, . . . , xn−1) . 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”.
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