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There is some classification of finite commutative local rings. For example how many not isomorphic finite local rings with the same order $p^k$ and the same residue field $\mathbb F_p$ exist?

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You surely do not mean finite field there. – Mariano Suárez-Alvarez Jun 5 '12 at 15:57
Miguel, you can edit the text of the question (by clicking on the edit link that appears right below it) It i better if you fix it there, and then we can remove these comments. – Mariano Suárez-Alvarez Jun 5 '12 at 16:16
Gracias Mariano – Miguel Jun 5 '12 at 17:10
Martin, I do not understand your comment, you want to explain? – Miguel Jun 6 '12 at 9:41
up vote 5 down vote accepted

Miguel, I S Cohen's structure theorems on complete local rings answers (more or less) your question, since a finite local ring is complete. Your ring $R$ must be a quotient of a ring of power series $k[[x_1,\dots, x_n]]$ where $k$ is either a finite field or the ring of Witt vectors over a finite field. You can take $n=m$ to be the minimal number of generators of the maximal ideal $\mathfrak{m}=(y_1,\dots,y_m)\subset R$, or $n=m-1$ if $R$ does not contain the residue field.

The answer to your specific question looks like more complicated in general, but you can try to play with the theorem to get an answer. For instance, the only examples with $p^2$ elements are simply $\mathbb{F}_p[t]/(t^2)$, $\mathbb{Z}/(p^2)$, and $\mathbb{F}_{p^2}$.

I. S. Cohen On the Structure and Ideal Theory of Complete Local Rings Transactions of the American Mathematical Society, Vol. 59, No. 1 (Jan., 1946), pp. 54-106

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Thanks, your answer is very useful for me. – Miguel Jun 6 '12 at 7:37
Good answer, but it seems to me that classifying those rings using just Cohen's theorem becomes very quickly intractable. I would like to hear about whatever is known on this classification beyond's Cohen's theorem. – Joël Jun 6 '12 at 16:58

Corbas and Williams answered this question for local rings with p^k elements, where k is at most 5. See "Rings of Order p^5 Part I. Nonlocal Rings" and "Rings of Order p^5 Part II. Local Rings". Both papers are in Journal of Algebra, vol. 231; the first on pages 677-690 and the second on pages 691-704.

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