## Small Rings

### Information

Thanks to students Kristine, Peter D, David H, Brandon, Nathan, Matt, Ben, Liz T, Serena, Elizabeth M, Peter Q, Jit, and Emily for their contributions! (All are in Math 322, the second semester of Abstract Algebra.) Thanks also to Professor Siehler for his valuable assistance in this work. This page is copyright 2005 by Gregory Dresden, of the Department of Mathematics here at Washington & Lee University.

Our goal is to find "nice" descriptions of small rings. Mathworld has a definition and discussion of rings, if you'd like a brief refresher. Note that our page is one of the references listed on the Mathworld site! We're also (for now at least) the first item returned by Google when searching for "Small Rings". (Here's a cache of Google's search results, in case Google changes its mind.)

We would like to represent each ring using only subsets of:
• matrices (over Z or Zn)
• modular rings Zn
• factor rings of Z[x]
• direct products of the above three types of rings
We are being fairly arbitrary with what is a "nice" description of a ring, but these seem to fit most people's description of "nice" rings. Notice also that the third item on the list can cover a lot of different cases; for example, the ring of size four given by Z2[i] is the factor ring Z[x]/<2,x2+1>.

For further study, here are some good references:
• Christof Nöbauer's web site contains some info on finite rings. Of particular interest are his technical report Numbers of small rings (ps-file, middle of the page) and this chart on the number of rings of prime-power order.
• Colin Fletcher's article Rings of small order (Mathematical Gazette, volume 64 [1980], no. 427, pages 9--22) can be found in our library. (Sorry, I can't find it online.)
• Benjamin Fine has a nice article, Classification of Finite Rings of Order p² (Mathematics Magazine, Vol. 66, No. 4 [Oct., 1993], pages 248--252) which can be downloaded for free from this JSTOR link so long as you're on a University computer. (If you're having trouble, the math/science library can help you out.)
• R. Raghavendran has a lovely article, Finite Associative Rings (Compositio Mathematica, Vol. 21, No. 2 [1969], pages 195-229), which covers much of the material later used by Nöbauer
• Eric Weisstein's Mathworld site was mentioned above, but it's worth repeating.
• The number of rings of size 0, 1, 2, 3, 4, 5, etc. forms the sequence 0, 1, 2, 2, 11, 2, etc., also known as sequence number A037234 from Neil Sloane's On-line Encyclopedia of Integer Sequences.
If you find any other good articles or references, let me know!

Here's what we have so far:

### Rings of Size 4

There are eleven rings of size 4, as follows:
• Three rings over Z4.
• Eight rings over Z2+Z2:
• Three commutative with unity.
• Three commutative without unity.
• Two non-commutative.
This was also the solution to problem E1648 in the MAA Monthly (Vol. 71, No. 8 [Oct., 1964], pages 918--920) by David Singmaster and D. M. Bloom, and can be found at this JSTOR link.

If you're interested, here are my old hand-written notes on rings of size 4.

### Rings of Size p

There are only two rings of size p (for p prime), as follows:
• The ring Zp (standard multiplication).
• The ring of size p with trivial multiplication, which can be represented as the subring <p> of the ring Zp2.

### Rings of Size pq

There are four rings of size pq (for p,q distinct prime). Any ring of size pq will have an ideal of size p and an ideal of size q, with trivial multiplication occurring between them. Thus, any ring of size pq can be written as a direct product of rings of prime powers; that is, the ring will have elements of form (a,b) where a is from a ring of size p, and b from a ring of size q.

### Rings of Size p2

Just as in the case of rings of size 4, there are eleven rings of size p2, as follows:
• Three rings over Zp2.
• Eight rings over Zp+Zp:
• Three commutative with unity.
• Three commutative without unity.
• Two non-commutative.
Benjamin Fine's article (discussed above) contains an abstract description of these eleven rings.

### Rings of Size p2q

Any ring of size p2q will be a direct product of two smaller rings with trivial multiplication occurring "between" the two rings. Thus, there will be twenty-two rings of size p2q.

### Rings of Size p3

There are 52 (if p=2) or 53 (if p>2) rings of size p3, as follows:
• Four rings over Zp3.
• Twenty (or twenty-one if p>2) rings over Zp2+Zp.
• Twenty-eight rings over Zp+Zp+Zp.
There are eleven (twelve if p>2) rings of size p3 with identity, broken down as follows:
• One over Zp3 (namely, the ring Zp3).
• Three (or four if p>2) over Zp2+Zp. According to R. Raghavendran's article, these rings are the following:
• Zp2+Zp with standard multiplication.
• Z[x]/<p2, px, x2>.
• Z[x]/<p2, px, x2 - p>.
• Z[x]/<p2, px, x2 - mp>, for m a quadratic non-residue mod p. (Note that this is only possible for p>2 !!)
• Seven over Zp+Zp+Zp.

Gregory Dresden, Department of Mathematics at Washington & Lee University