Classification of finite commutative rings

Question

Is there a classification of finite commutative rings available? If not, what are the best structure theorem that are known at present? All I know is a result that every finite commutative ring is a direct product of local commutative rings (this is correct, right?) in some paper which computes the size of the general linear group over that ring.

Answer 1

Yes, a finite ring $R$ is a finite direct sum of local finite rings. As a first step, for each prime $p$ there is a subring $R_p$ of $R$ corresponding to the elements annihilated by the powers of $p$. $\require{enclose} \enclose{horizontalstrike}{R_p\ \style{font-family:inherit;}{\text{is then an}}\hspace{-7mm}}$ $\enclose{horizontalstrike}{\style{font-family:inherit;}{\text{algebra over}}\ \mathbb{Z}/p.}$ $R_p$ then resembles an algebra over $\mathbb{Z}/p$ and it could be one, but it can also have a more complicated structure as an abelian $p$-group (see below). This step generalizes to maximal ideals: For each maximal ideal $m$, $R_m$ is the subring of elements annihilated by $m^n$ for some $n$, and $R$ is the direct sum of these subrings, which are local rings.

It is not difficult to write down a rough partial classification of of local finite rings. If $R$ is local with maximal ideal $m$, it is resembles an algebra over the finite field $F = R/m$; the associated graded ring is such an algebra. If you choose a basis $x_1,\ldots,x_n$ for $m/m^2$, then $R$ or its associated graded is a quotient of the polynomial ring $F[\vec{x}]$ in which only finitely many monomials are non-zero. You can make a diagram of these non-zero monomials; they can be any order ideal in the $n$-dimensional orthant. Or, in basis-independent form, $R$ has a length, which is the largest nonvanishing power of $m$, and each $m^k/m^{k+1}$ is some quotient of the $k$th symmetric power of the generating vector space $V = m/m^2$.

After that, the non-zero monomials may be linearly dependent (and never mind that $R$ might be more complicated than its associated graded). Informally, there will be an endless stream of partial results and there will never be a complete classification when the length of the local ring is 3 or more. To see this, suppose that $m^4 = 0$, and suppose that $m^3$ is only one dimension shy of $S^3(V)$. Then the ring is defined by an arbitrary symmetric trilinear form in $V$. These make a "wild" sequence of algebraic varieties, in the same sense that people say that the representation theories of certain rings are wild. For instance, I think (not quite sure) that it is NP-hard to determine when two such trilinear forms are equivalent. NP-hardness is not by itself rigorously equivalent to no classification, but informally the classification is an intractable mess.

If the nonvanishing monomials in $R$ are linearly independent, then it is a toric local ring. Toric local rings are certainly a tractable class of finite rings.

The situation is similar to non-commutative $p$-groups, which are also wild and will never be classified. In both cases, certain classes have a nice structure. It is also interesting to make estimates for how many there are.

Note: Corrected per comment.

Answer 2

This is a very interesting question related to the Hilbert scheme $Hilb^n(\mathbb A^d)$ classifying $n$ points in affine space $\mathbb A^d$. I don't think there is a classification but there is an estimate for the number of commutative rings of order $\leq N$. It is

$$exp[\frac{2}{27} \frac{log(N)^3}{(log 2)^2} \; +O(log(N)^{\frac {8}{3}})] \quad for N\to \infty $$

The proof of this result due to Bjorn Poonen and of many related interesting theorems is in his article

You will also find astonishing conjectures in the article like:

The fraction of local rings of order $\leq N$ among all commutative rings $A$ of order $\leq N$ tends to 1. Same limit 1 for the fraction of rings "of characteristic 8" in the sense that $8 . 1_A =0$ but $4 .1 _A \neq 0$.

Answer 3

The characterization of Artinian rings is relevant of course. See also the book "Finite commutative rings and their applications" and this web page.

Answer 4

As always one should check out the OEIS for questions of this type. In this case see http://oeis.org/A027623