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

We would like to represent each ring using only subsets of:

- matrices (over
or**Z**)**Z**_{n} - modular rings
**Z**_{n} - factor rings of
**Z**[x] - direct products of the above three types of rings

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.

Here's what we have so far:

- Three rings over
.**Z**_{4} - Eight rings over
:**Z**_{2}+**Z**_{2}- Three commutative with unity.
- Three commutative without unity.
- Two non-commutative.

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

- The ring
(standard multiplication).**Z**_{p} - The ring of size
`p`with trivial multiplication, which can be represented as the subring`<p>`of the ring.**Z**_{p2}

- Three rings over
.**Z**_{p2} - Eight rings over
:**Z**_{p}+**Z**_{p}- Three commutative with unity.
- Three commutative without unity.
- Two non-commutative.

- Four rings over
.**Z**_{p3} - Twenty (or twenty-one if
`p>2`) rings over.**Z**_{p2}+**Z**_{p} - Twenty-eight rings over
.**Z**_{p}+**Z**_{p}+**Z**_{p}

- One over
(namely, the ring**Z**_{p3}).**Z**_{p3} - Three (or four if
`p>2`) over. According to R. Raghavendran's article, these rings are the following:**Z**_{p2}+**Z**_{p}with standard multiplication.**Z**_{p2}+**Z**_{p}^{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`.**Z**_{p}+**Z**_{p}+**Z**_{p}

Gregory Dresden, Department of Mathematics at Washington & Lee University