TITLE: There are at least three mutually non-isomorphic rings with $4$ elements?
QUESTION [13 upvotes]: Is the following statement is true?

There are at least three mutually non-isomorphic rings with $4$ elements.

I have no idea or counterexample at the moment. Please help. So far I know about that a group of order $4$ is abelian and there are two non isomorphic groups of order $4$ like $K_4(non cyclic)$ and $\mathbb Z_4(cyclic)$.

REPLY [3 votes]: These can be exhaustively enumerated using alg.  To enumerate the number of non-isomorphic rings of orders $1,2,\ldots,8$, we enter:
./alg theories/ring.th --size 1-8 --count

which outputs:
# Theory ring

    Constant 0.
    Unary ~.
    Binary + *.

    Axiom plus_commutative: x + y = y + x.
    Axiom plus_associative: (x + y) + z = x + (y + z).
    Axiom zero_neutral_left: 0 + x = x.
    Axiom zero_neutral_right: x + 0 = x.
    Axiom negative_inverse: x + ~ x = 0.
    Axiom negative_inverse: ~ x + x = 0.
    Axiom zero_inverse: ~ 0 = 0.
    Axiom inverse_involution: ~ (~ x) = x.

    Axiom mult_associative: (x * y) * z = x * (y * z).
    Axiom distrutivity_right: (x + y) * z = x * z + y * z.
    Axiom distributivity_left: x * (y + z) = x * y + x * z.

    size | count
    -----|------
       1 | 1
       2 | 2
       3 | 2
       4 | 11
       5 | 2
       6 | 4
       7 | 2
       8 | 52

Check the numbers [2, 2, 11, 2, 4, 2, 52](http://oeis.org/search?q=2,2,11,2,4,2,52) on-line at oeis.org

If we want it to print out the addition and multiplication tables, we can remove --count from the command line:
./alg theories/ring.th --size 4

If we want to work with unital rings, we can use:
./alg theories/unital_ring.th --size 1-8 --count

which gives:
# Theory unital_ring

    Theory unital_ring.

    Constant 0 1.
    Unary ~.
    Binary + *.

    Axiom plus_commutative: x + y = y + x.
    Axiom plus_associative: (x + y) + z = x + (y + z).
    Axiom zero_neutral_left: 0 + x = x.
    Axiom negative_inverse: x + ~ x = 0.
    Axiom mult_associative: (x * y) * z = x * (y * z).
    Axiom one_unit_left: 1 * x = x.
    Axiom one_unit_right: x * 1 = x.
    Axiom distrutivity_right: (x + y) * z = x * z + y * z.
    Axiom distributivity_left: x * (y + z) = x * y + x * z.

    # Consequences of axioms that make alg run faster:

    Axiom zero_neutral_right: x + 0 = x.
    Axiom negative_inverse: ~ x + x = 0.
    Axiom zero_inverse: ~ 0 = 0.
    Axiom inverse_involution: ~ (~ x) = x.
    Axiom mult_zero_left: 0 * x = 0.
    Axiom mult_zero_right: x * 0 = 0.


    size | count
    -----|------
       1 | 0
       2 | 1
       3 | 1
       4 | 4
       5 | 1
       6 | 1
       7 | 1
       8 | 11

Check the numbers [1, 1, 4, 1, 1, 1, 11](http://oeis.org/search?q=1,1,4,1,1,1,11) on-line at oeis.org