NUMBERS OF FINITE RINGS

ACHIM FLAMMENKAMP & CHRISTOF NÖBAUER

1. Finite Rings

There appears to be quite an interest in the spectra of small rings; we include a survey on the theoretical work of this topic that we are aware of as well as some results of our own computational efforts. Apart from our own work, we are aware of the work by Michael Lin ([Lin92]).

1.1. Spectra of rings. The textbook [LW80] contains an overview on number theorems for small rings on pages 133 - 141; we quote the main theorems and sources (always let m, n ∈ ℕ and p a prime):

First, we notice that it suffices to consider rings of prime power order:

Theorem 1. [Sho30] Every finite ring R can be uniquely (up to isomorphism) decomposed into a direct sum of rings of prime power order.

Theorem 2. [Bea48] There are n+1 non-isomorphic rings with additive group ℤ_{p^{n}} and all are commutative. Let a be any generator of ℤ_{p^{n}}. Then, for each integer i with 0 ≤ i ≤ n, there is a ring with a^{p^{i}} = a. If i > 0, then i is smallest positive, otherwise (i = 0) a is nilpotent and we have a^{2} = 0. Only if i = n, the ring has an identity.

Theorem 3. [Sco22, BS49] There are 8 non-isomorphic rings with additive group ℤ_{p}^{2} ([LW80] gives an explicit description of these rings). Let a and b be generators of ℤ_{p}^{2} — then there are the possibilities:

R a^{2} ab ba b^{2} comm. e  remark
T_{1} 0 0 0 0  ×  —  R ≅ R_{0}⊕R_{0}
T_{2} b  0 0 0  ×  —
T_{3} a 0 0 0  ×  —  R ≅ R_{1}⊕R_{0}
T_{4} a  b 0 0  —  —
T_{5} a 0 b 0  —  —
T_{6} a  b b 0  ×  a
T_{7} a 0 0 b  × a+b  R ≅ R_{1}⊕R_{1}
T_{8a} a+b a a b  ×  b R ≅ 𝔽_{4}  p = 2
T_{8b} μb a a b  ×  b R ≅ 𝔽_{p^{2}}  p ≠ 2

μ ∈ ℤ_{p} is any fixed quadratic non-residue modulo p.

From these theorems, it follows that there are 3 + 8 = 11 nonisomorphic rings of order p^{2}, 9 of these are commutative, 4 have an identity and 3 (S_{0}, T_{1}, T_{2}) are nilpotent.

2 ACHIM FLAMMENKAMP & CHRISTOF NÖBAUER

Theorem 4. [FW74, LW80] There are (3p+4)n−4p+11 non-isomorph-ic rings with additive group ℤ_{p^{n}} × ℤ_{p} (n ≥ 2) if p is an odd prime and 9n + 2 if p = 2.

Theorem 5. [FW74, LW80] There are p + 27 non-isomorphic rings with additive group ℤ_{p}^{3} if p is odd and 28 if p = 2.

These two theorems give a total of 3p + 50 non-isomorphic rings of order p^{3} for odd p and 52 for p = 2. This result is also obtained by [AE82], where an explicit description of all sub-cases is given. Furthermore, we have

Theorem 6. [LW80, AE82] The number of non-isomorphic non-com-mutative rings with

• additive group ℤ_{p}^{3} is 12 for p = 2 and p + 11 if p an odd prime.

• additive group ℤ_{p} ×ℤ_{p}^{2} is 6 for p = 2 and 2p+3 if p an odd prime.

1.2. Spectra of rings with identity.

Theorem 7. [Rag69] There are 11 rings with identity of order 8 and 12 rings with identity of order p^{3} (p an odd prime).

Theorem 8. [Rag69] For all rings of order p^{3} with p a prime, there exists exactly one non-commutative ring with identity. Its additive group is isomorphic to ℤ_{p}^{3}.

Theorem 9. [Wie74] If p is an odd prime, m ≥ 1, n ≥ 0, r = ⌊\frac{n}{2}⌋, and s = ⌊\frac{m}{2}⌋ then there are

N = \{ \begin{matrix}(m−2r+1)p^{r} + \frac{p^{n−r}−1}{p−1} + 3\frac{p^{r}−1}{p−1} + m & if n ≤ m \\ (m−2s+1)p^{s} + \frac{p^{m−s}−1}{p−1} + 3\frac{p^{s}−1}{p−1} + m & if n > m \end{matrix}

non-isomorphic rings with identity and additive group ℤ_{p^{m+n}} × ℤ_{p^{m}}.

Theorem 10. [Wie85] If m ≥ 1, n ≥ 0, r = ⌊\frac{n}{2}⌋, and s = ⌊\frac{m}{2}⌋ then there are

N = \{ \begin{matrix}4m + 3n − 2 & if n ≤ m−2 and n ≤ 1 \\ 2^{n−r+1} + (m−2r+3)2^{r+1} + 4m−17 & if 2 ≤ n ≤ m−2 \\ 4m − 1 & if n > m−2 and m ≤ 2 \\ 2^{m−s+1} + (m−2s+5)2^{s} + 4m−17 & if n > m−2 > 0 \end{matrix}

non-isomorphic rings with identity and additive group ℤ_{2^{m+n}} × ℤ_{2^{m}}.

1.3. Summary. These theorems allow us to determine the numbers of small rings (order < 100) as long as their order is not divisible by a fourth prime power – for rings of order p^{4}, we are only aware of partial solutions, e.g. [KP69, Flo73, Wie88].

The results on the numbers of total rings and commutative rings of order 16, 32, 64 and 81 whose additive group is not cyclic ( ℤ_{2}^{2}×ℤ_{4}, ℤ_{4}^{2}, ℤ_{2} × ℤ_{16}, ℤ_{4} × ℤ_{8}, ℤ_{2}^{2} × ℤ_{8}, ℤ_{2} × ℤ_{4}^{2}, ℤ_{2} × ℤ_{32}, ℤ_{4} × ℤ_{16}, ℤ_{8}^{2}, ℤ_{2}^{2} × ℤ_{16}, ℤ_{2} × ℤ_{4} × ℤ_{8}, ℤ_{3} × ℤ_{27}, ℤ_{9}^{2}) was obtained by computer calculation.

NUMBERS OF FINITE RINGS 3

additive group # total rings

(with e)

# commutative (with e)

ℤ_{2}  2  1  2  1
ℤ_{3}  2  1  2  1
ℤ_{4}  3  1  3  1
ℤ_{2}^{2}  8  3  6  3
ℤ_{5}  2  1  2  1
ℤ_{2} × ℤ_{3}  4  1  4  1
ℤ_{7}  2  1  2  1
ℤ_{8}  4  1  4  1
ℤ_{2} × ℤ_{4}  20  3  14  3
ℤ_{2}^{3}  28  7  16  6
ℤ_{9}  3  1  3  1
ℤ_{3}^{2}  8  3  6  3
ℤ_{2} × ℤ_{5}  4  1  4  1
ℤ_{11}  2  1  2  1
ℤ_{3} × ℤ_{4}  6  1  6  1
ℤ_{2}^{2} × ℤ_{3}  16  3  12  3
ℤ_{13}  2  1  2  1
ℤ_{2} × ℤ_{7}  4  1  4  1
ℤ_{3} × ℤ_{5}  4  1  4  1
ℤ_{16}  5  1  5  1
ℤ_{2} × ℤ_{8}  29  3  20  3
ℤ_{4}^{2}  66  6  28  6
ℤ_{2}^{2} × ℤ_{4}  170  15  65  11
ℤ_{2}^{4}  25  44  16
ℤ_{17}  2  1  2  1
ℤ_{2} × ℤ_{9}  6  1  6  1
ℤ_{2} × ℤ_{3}^{2}  16  3  12  3
ℤ_{19}  2  1  2  1
ℤ_{4} × ℤ_{5}  6  1  6  1
ℤ_{2}^{2} × ℤ_{5}  16  3  12  3
ℤ_{3} × ℤ_{7}  4  1  4  1
ℤ_{2} × ℤ_{11}  4  1  4  1
ℤ_{23}  2  1  2  1
ℤ_{3} × ℤ_{8}  8  1  8  1
ℤ_{2} × ℤ_{3} × ℤ_{4}  40  3  28  3
ℤ_{2}^{3} × ℤ_{3}  56  7  32  6
ℤ_{25}  3  1  3  1
ℤ_{5}^{2}  8  3  6  3
ℤ_{2} × ℤ_{13}  4  1  4  1
ℤ_{27}  4  1  4  1
ℤ_{3} × ℤ_{9}  25  4  16  4
ℤ_{3}^{3}  30  7  16  6
ℤ_{4} × ℤ_{7}  6  1  6  1
ℤ_{2}^{2} × ℤ_{7}  16  3  12  3

Table 1. Numbers of non-isomorphic rings with 2 ≤ order ≤ 28.

4 ACHIM FLAMMENKAMP & CHRISTOF NÖBAUER

additive group # total rings

(with e)

# commutative (with e)

ℤ_{29}  2  1  2  1
ℤ_{2} × ℤ_{3} × ℤ_{5}  8  1  8  1
ℤ_{31}  2  1  2  1
ℤ_{32}  6  1  6  1
ℤ_{2} × ℤ_{16}  38  3  26  3
ℤ_{4} × ℤ_{8}  167  7  64  7
ℤ_{2}^{2} × ℤ_{8}  251  15  95  11
ℤ_{2} × ℤ_{4}^{2}  11593  24  340  17
ℤ_{2}^{3} × ℤ_{4}  79  345  36

ℤ_{2}^{5}

ℤ_{3} × ℤ_{11}  4  1  4  1
ℤ_{2} × ℤ_{17}  4  1  4  1
ℤ_{5} × ℤ_{7}  4  1  4  1
ℤ_{4} × ℤ_{9}  9  1  9  1
ℤ_{2}^{2} × ℤ_{9}  24  3  18  3
ℤ_{3}^{2} × ℤ_{4}  24  3  18  3
ℤ_{2}^{2} × ℤ_{3}^{2}  64  9  36  9
ℤ_{37}  2  1  2  1
ℤ_{2} × ℤ_{19}  4  1  4  1
ℤ_{3} × ℤ_{13}  4  1  4  1
ℤ_{5} × ℤ_{8}  8  1  8  1
ℤ_{2} × ℤ_{4} × ℤ_{5}  40  3  28  3
ℤ_{2}^{3} × ℤ_{5}  56  7  32  6
ℤ_{41}  2  1  2  1
ℤ_{2} × ℤ_{3} × ℤ_{7}  8  1  8  1
ℤ_{43}  2  1  2  1
ℤ_{4} × ℤ_{11}  6  1  6  1
ℤ_{2}^{2} × ℤ_{11}  16  3  12  3
ℤ_{5} × ℤ_{9}  6  1  6  1
ℤ_{3}^{2} × ℤ_{5}  16  3  12  3
ℤ_{2} × ℤ_{23}  4  1  4  1
ℤ_{47}  2  1  2  1
ℤ_{3} × ℤ_{16}  10  1  10  1
ℤ_{2} × ℤ_{3} × ℤ_{8}  58  3  40  3
ℤ_{3} × ℤ_{4}^{2}  132  6  56  6
ℤ_{2}^{2} × ℤ_{3} × ℤ_{4}  340  15  130  11
ℤ_{2}^{4} × ℤ_{3}  25  88  16
ℤ_{49}  3  1  3  1
ℤ_{7}^{2}  8  3  6  3
ℤ_{2} × ℤ_{25}  6  1  6  1
ℤ_{2} × ℤ_{5}^{2}  16  3  12  3
ℤ_{3} × ℤ_{17}  4  1  4  1
ℤ_{4} × ℤ_{13}  6  1  6  1
ℤ_{2}^{2} × ℤ_{13}  16  3  12  3

Table 2. Numbers of non-isomorphic rings with 29 ≤ order ≤ 52.

NUMBERS OF FINITE RINGS 5

additive group # total rings

(with e)

# commutative (with e)

ℤ_{53}  2  1  2  1
ℤ_{2} × ℤ_{27}  8  1  8  1
ℤ_{2} × ℤ_{3} × ℤ_{9}  50  4  36  4
ℤ_{2} × ℤ_{3}^{3}  60  7  32  6
ℤ_{5} × ℤ_{11}  4  1  4  1
ℤ_{7} × ℤ_{8}  8  1  8  1
ℤ_{2} × ℤ_{4} × ℤ_{7}  40  3  28  3
ℤ_{2}^{3} × ℤ_{7}  56  7  32  6
ℤ_{2} × ℤ_{19}  4  1  4  1
ℤ_{59}  2  1  2  1
ℤ_{3} × ℤ_{4} × ℤ_{5}  12  1  12  1
ℤ_{2}^{2} × ℤ_{3} × ℤ_{5}  32  3  24  3
ℤ_{61}  2  1  2  1
ℤ_{2} × ℤ_{31}  4  1  4  1
ℤ_{7} × ℤ_{9}  6  1  6  1
ℤ_{3}^{2} × ℤ_{7}  16  3  12  3
ℤ_{64}  7  1  7  1
ℤ_{2} × ℤ_{32}  47  3  32  3
ℤ_{4} × ℤ_{16}  248  7  95  7
ℤ_{8}^{2}  301  10  79  10
ℤ_{2}^{2} × ℤ_{16}  332  15  125  11
ℤ_{2} × ℤ_{4} × ℤ_{8}  27332  105  999  46
ℤ_{4}^{3}  801559  27  1839  16
ℤ_{2}^{3} × ℤ_{8}  79  520  36
ℤ_{2}^{2} × ℤ_{4}^{2}  232  98

ℤ_{2}^{4} × ℤ_{4} ℤ_{2}^{6}

ℤ_{5} × ℤ_{13}  4  1  4  1
ℤ_{2} × ℤ_{3} × ℤ_{11}  8  1  8  1
ℤ_{67}  2  1  2  1
ℤ_{4} × ℤ_{17}  6  1  6  1
ℤ_{2}^{2} × ℤ_{17}  16  3  12  3
ℤ_{3} × ℤ_{23}  4  1  4  1
ℤ_{2} × ℤ_{5} × ℤ_{7}  8  1  8  1
ℤ_{71}  2  1  2  1
ℤ_{8} × ℤ_{9}  12  1  12  1
ℤ_{3}^{2} × ℤ_{8}  32  7  24  7
ℤ_{2} × ℤ_{4} × ℤ_{9}  60  3  42  3
ℤ_{2}^{3} × ℤ_{9}  84  7  48  7
ℤ_{2} × ℤ_{3}^{2} × ℤ_{4}  160  9  104  9
ℤ_{2}^{3} × ℤ_{3}^{2}  224  21  96  21
ℤ_{73}  2  1  2  1
ℤ_{2} × ℤ_{37}  4  1  4  1
ℤ_{3} × ℤ_{25}  6  1  6  1
ℤ_{3} × ℤ_{5}^{2}  16  3  12  3

Table 3. Numbers of non-isomorphic rings with 53 ≤ order ≤ 75.

6 ACHIM FLAMMENKAMP & CHRISTOF NÖBAUER

additive group # total rings

(with e)

# commutative (with e)

ℤ_{4} × ℤ_{19}  6  1  6  1
ℤ_{2}^{2} × ℤ_{19}  16  3  12  3
ℤ_{7} × ℤ_{11}  4  1  4  1
ℤ_{2} × ℤ_{3} × ℤ_{13}  8  1  8  1
ℤ_{5} × ℤ_{16}  10  1  10  1
ℤ_{2} × ℤ_{5} × ℤ_{8}  58  3  40  3
ℤ_{4}^{2} × ℤ_{5}  132  6  56  6
ℤ_{2}^{2} × ℤ_{4} × ℤ_{5}  340  15  130  11
ℤ_{2}^{4} × ℤ_{5}  25  88  16
ℤ_{81}  5  1  5  1
ℤ_{3} × ℤ_{27}  38  4  24  4
ℤ_{9}^{2}  175  5  35  5
ℤ_{3}^{2} × ℤ_{9}  264  20  73  13

ℤ_{3}^{4}

ℤ_{2} × ℤ_{41}  4  1  4  1
ℤ_{83}  2  1  2  1
ℤ_{3} × ℤ_{4} × ℤ_{7}  12  1  12  1
ℤ_{2}^{2} × ℤ_{3} × ℤ_{7}  32  3  24  3
ℤ_{5} × ℤ_{17}  4  1  4  1
ℤ_{2} × ℤ_{43}  4  1  4  1
ℤ_{3} × ℤ_{29}  4  1  4  1
ℤ_{8} × ℤ_{11}  8  1  8  1
ℤ_{2} × ℤ_{4} × ℤ_{11}  40  3  28  3
ℤ_{2}^{3} × ℤ_{11}  56  7  32  6
ℤ_{89}  2  1  2  1
ℤ_{2} × ℤ_{5} × ℤ_{9}  12  1  12  1
ℤ_{2} × ℤ_{3}^{2} × ℤ_{5}  32  3  24  3
ℤ_{7} × ℤ_{13}  4  1  4  1
ℤ_{4} × ℤ_{23}  6  1  6  1
ℤ_{2}^{2} × ℤ_{23}  16  3  12  3
ℤ_{3} × ℤ_{31}  4  1  4  1
ℤ_{2} × ℤ_{47}  4  1  4  1
ℤ_{5} × ℤ_{19}  4  1  4  1
ℤ_{3} × ℤ_{32}  12  1  12  1
ℤ_{2} × ℤ_{3} × ℤ_{16}  76  3  52  3
ℤ_{3} × ℤ_{4} × ℤ_{8}  334  7  128  7
ℤ_{2}^{2} × ℤ_{3} × ℤ_{8}  502  15  184  11
ℤ_{2} × ℤ_{3} × ℤ_{4}^{2}  23186  24  680  17
ℤ_{2}^{3} × ℤ_{3} × ℤ_{4}  79  690  36

ℤ_{2}^{5} × ℤ_{3}

ℤ_{97}  2  1  2  1
ℤ_{2} × ℤ_{49}  6  1  6  1
ℤ_{2} × ℤ_{7}^{2}  16  3  12  3
ℤ_{9} × ℤ_{11}  6  1  6  1
ℤ_{3}^{2} × ℤ_{11}  16  3  12  3

Table 4. Numbers of non-isomorphic rings with 76 ≤ order ≤ 99.

1.4. Classification of the finite rings of order p^{3}.

If the additive group of the ring equals ℤ_{p^{3}}, then, according to Theorem 2, there are the four rings a^{2} = p^{j}a with 0 ≤ j ≤ 3 and a a generator of the ring. Notice, i of Theorem 2 is here n − j.

Now we mainly quote the results of Antipkin and Elizarov [AE82].

If the additive group of the ring equals ℤ_{p} × ℤ_{p^{2}}, then let a and b be generators of the ring with ord a = p and ord b = p^{2}.

There are 2 · 3 rings which are a direct sum of two rings:

no. a^{2} ab ba b^{2} comm. e  remark
1 0 0 0 0  ×  — R ≅ R_{0} ⊕ S_{0}
2 a 0 0 0  ×  — R ≅ R_{1} ⊕ S_{0}
3 0 0 0 pb  ×  — R ≅ R_{0} ⊕ S_{1}
4 a 0 0 pb  ×  — R ≅ R_{1} ⊕ S_{1}
5 0 0 0 b  ×  — R ≅ R_{0} ⊕ S_{2}
6 a 0 0 b  × a+b R ≅ R_{1} ⊕ S_{2}

S_{0},S_{1} and S_{2} denote the three rings with additive group ℤ_{p^{2}}. R_{0} and R_{1} denote the two rings with additive group ℤ_{p}.

For the remaining indecomposable rings, we have:

no. a^{2} ab ba b^{2} comm. e remark
7 0 0 pb 0  — —
8 0 pb 0 0  — —
9a 0 pb pb 0  × —  p = 2
9b 0 pb −pb 0  — —  p ≠ 2
10 0 0 0 a  × —
11 0 pb pb a  × —
12 0 pb pb μa × —  p ≠ 2
13a 0 pb αpb pb × — α = 1
13b 0 pb αpb pb — — ∀ α ∈ ℤ_{p}\{0, 1}
14 pb 0 0 0  × —
15 pb pb 0 0  — —
16 pb 0 0 pb  × —
17 μpb 0 0 pb × — p ≠ 2
18 αpb 0 pb pb — —  ∀ α ∈ ℤ_{p}\{0}
19 0 a 0 b  — —
20 0 0 a b  — —
21 0 a a b  ×  b
22 pb a a b  ×  b
23 μpb a a b  ×  b  p ≠ 2

μ ∈ ℤ_{p} is any fixed quadratic non-residue modulo p.

8 ACHIM FLAMMENKAMP & CHRISTOF NÖBAUER

If the additive group of the ring equals ℤ_{p} × ℤ_{p} × ℤ_{p}, then let a, b and c be generators of the ring.

There are 2 · 8 − 2 rings which are a direct sum of two or three rings:

no. a^{2} ab ba b^{2} ac ca bc cb c^{2} comm.  e  remark
1 0  0 0 0 0 0 0 0 0  ×  — R ≅ R_{0}⊕R_{0}⊕R_{0}
2 b  0 0 0 0 0 0 0 0  ×  —  R ≅ R_{0}⊕T_{2}
3 a  0 0 0 0 0 0 0 0  ×  — R ≅ R_{0}⊕R_{0}⊕R_{1}
4 a  b 0 0 0 0 0 0 0 —  —  R ≅ R_{0}⊕T_{4}
5 a  0 b 0 0 0 0 0 0 —  —  R ≅ R_{0}⊕T_{5}
6 a  b b 0 0 0 0 0 0  ×  —  R ≅ R_{0}⊕T_{6}
7 a  0 0 b 0 0 0 0 0  ×  — R ≅ R_{0}⊕R_{1}⊕R_{1}
8a a + b a a b 0 0 0 0 0  ×  —  R ≅ R_{0}⊕T_{8a}
8b μb a a b 0 0 0 0 0  ×  —  R ≅ R_{0}⊕T_{8b}
9 b  0 0 0 0 0 0 0 c  ×  —  R ≅ R_{1}⊕T_{2}
10 a  b 0 0 0 0 0 0 c —  —  R ≅ R_{1}⊕T_{4}
11 a  0 b 0 0 0 0 0 c —  —  R ≅ R_{1}⊕T_{5}
12 a  b b 0 0 0 0 0 c  ×  a+c  R ≅ R_{1}⊕T_{6}
13 a  0 0 b 0 0 0 0 c  × a+b+c R ≅ R_{1}⊕R_{1}⊕R_{1}
14a a + b a a b 0 0 0 0 c  ×  b+c  R ≅ R_{1}⊕T_{8a}
14b μb a a b 0 0 0 0 c  ×  b+c  R ≅ R_{1}⊕T_{8b}

μ ∈ ℤ_{p} is any fixed quadratic non-residue modulo p.

T_{k} denote the rings with additive group ℤ_{p} × ℤ_{p} for 1 ≤ k ≤ 8.

R_{0} and R_{1} denote the two rings with additive group ℤ_{p}.

For the remaining indecomposable rings, we have:

no. a^{2} ab ba b^{2} ac ca bc cb c^{2} comm. e remark
15 a b b c c c νa + b νa + b νb + c ×  a R ≅ 𝔽_{p^{3}}
16 a 0 0 b c 0  0  c  0  — a+b
17 a b b c c c  0  0  0  ×  a
18 a b b 0 c c  0  0  0  ×  a
19 a 0 b 0 0 c  0  0  0  —  —
20 a 0 0 0 0 c  0  0  0  —  —
21 a b b 0 c 0  0  0  0  —  —
22 a 0 b 0 c 0  0  0  0  —  —
23 a b 0 0 c 0  0  0  0  —  —
24 b c c 0 0 0  0  0  0  ×  —
25a 0 c c 0 0 0  0  0  0  ×  —  p = 2
25b 0 c −c 0 0 0  0  0  0  —  —  p ≠ 2
26 c 0 0 c 0 0  0  0  0  ×  —
27 c 0 0 μc 0 0  0  0  0  ×  —  p ≠ 2
28 c c 0 αc 0 0  0  0  0  —  — ∀ α ∈ ℤ_{p}

μ ∈ ℤ_{p} is any fixed quadratic non-residue modulo p. ν ∈ ℤ_{p} : x^{3} − x − ν is non-factorizable modulo p. add. group # total rings

(with e)

# commutative (with e)

{0}  1  0  1  0
ℤ_{p}  2  1  2  1
ℤ_{p^{2}}  3  1  3  1
ℤ_{p}^{2}  5  2  3  2
ℤ_{p^{3}}  4  1  4  1
ℤ_{p} × ℤ_{p^{2}} 2p+3(p%2)+10 p%2+2  2(p%2)+8 p%2+2
ℤ_{p}^{3}  p + p%2 + 12  4  7−p%2  3
ℤ_{p^{n}}  n + 1  1  n+1  1

Table 5. Number of non-isomorphic indecomposable rings.

References

[AE82] V. G. Antipkin and V. P. Elizarov, Rings of order p^{3}, Sib. Math. Journal 23 (1982), 457–464.

[Bea48] R. A. Beaumont, Rings with additive group which is the direct product of cyclic groups, Duke Mathematical Journal 15 (1948), 367–396.

[BS49] R. Ballieu and M. J. Schuind, Anneaux finis à module de type (p, p^{2}), Ann. Soc. Sci. Bruxelles 63 (1949), 11–23.

[Flo73] W. Flor, Über die Ringe der Ordnung p^{4}, PhD Dissertation, Technische Universität Wien, 1973.

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Email: achim@uni-bielefeld.de, christof.noebauer@algebra.uni-linz.ac.at

0 last modified: 1998-09-12