Does anyone know of a listing of all rings (or at least those that are commutative) of a given order? In particular $p^3$, for a prime $p$.
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Since a ring must be an abelian group, start by looking at all abelian group structures of order $p^3$. – walkar Aug 27 '15 at 13:25
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1It is not difficult to find all abelian groups of a given finite order. If the order you want is $p^3$, then there are $3$ distinct abelian groups (up to isomorphism). In each case, there are at least $2$ associative, commutative and distributive operations you can consider for multiplication. One is element-wise multiplication, and one is the trivial multiplication (all products are $0$). I don't know whether there are more. – Arthur Aug 27 '15 at 13:25
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@Arthur there are more, for instance, the field structure $\mathbb{F}_{p^3}$ – hunter Aug 27 '15 at 13:42
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do we require rings to have $1$? – hunter Aug 27 '15 at 13:45
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For the commutative case, I would guess that taking direct products of $\mathbb{Z}/p^i$ with finite fields $\mathbb{F}_{p^j}$ would account for all of them but this is just a guess. – Matt B Aug 27 '15 at 13:46
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@MattB consider $R= \mathbb{F_3}[X]/(X^3 - 1)$. It isn't a field, but it has a unique prime ideal so it can't be a non-trivial product either. – hunter Aug 27 '15 at 13:47
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5See this: http://math.stackexchange.com/questions/368323/structure-theorem-of-finite-rings – Sungjin Kim Aug 27 '15 at 14:01