A037291 -id:A037291 - cp's OEIS frontend

A027623 a(0) = 1; for n > 0, a(n) = number of rings with n elements.

1, 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, 390, 2, 22, 2, 22, 4, 4, 2, 104, 11, 4, 59, 22, 2, 8, 2

Offset: 0

Here a ring means (R,+,*): (R,+) is an abelian group, * is associative, a*(b+c) = a*b + a*c, (a+b)*c = a*c + b*c. Need not contain "1", * need not be commutative.
The sequence continues a(32) = ? (>18590), a(33) = 4, 4, 4, 121, 2, 4, 4, 104, 2, 8, 2, 22, 22, 4, 2, 780, 11, 22, 4, 22, 2, 118, 4, 104, 4, 4, 2, 44, 2, 4, 22 = a(63), a(64) = ? (> 829826). - Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000
The paper by Antipkin/Elizarov also gives the number a(p^3) of rings of order p^3. - Hans H. Storrer (storrer(AT)math.unizh.ch), Sep 16 2003
If n is a squared prime, there are 11 mutually nonisomorphic rings of order n [see Raghavendran, p. 228]. - R. J. Mathar, Apr 20 2008

			The 11 rings of order 4 (from _Christian G. Bower_):
  over C4: 1*1 = 0, 1 or 2;
  over C2 X C2 = <1> X <2>: (1*1,1*2,2*1,2*2) = 0000, 0001, 0002, 0012, 0102, 0112, 1002 or 1223.

V. G. Antipkin and V. P. Elizarov, Rings of order p^3, Sib. Math. J. vol 23 no 4 (1982) pp 457-464, MR0668331 (84d:16025).
R. Ballieu [ Math. Rev. 0022841; see also Math. Rev. 51#5655] showed a(8) = 52, a(p^3) = 3p + 50 if p is an odd prime.
Grigore Călugăreanu, Rings with very few nilpotents, An. Sţiinţ. Univ. Al. I. Cuza Iaşi. Mat. (2018), p. 149.
C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64 (1980) p. 13, 1980, see esp. p. 21.
Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
A. V. Lelechenko, Parity of the number of primes in a given interval and algorithms of the sublinear summation, arXiv preprint arXiv:1305.1639 [math.NT], 2013.
Desmond MacHale, Are There More Finite Rings than Finite Groups, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.
C. Noebauer, The Numbers of Small Rings.
C. Noebauer, Thesis on the enumeration of near-rings.
Christof Noebauer, The Numbers of Small Rings (PostScript).
R. Raghavendran, Finite associative rings, Compositio Mathematica vol 21 no 2 (1969) pp. 195-229.
Eric Weisstein's World of Mathematics, Ring.
Index to sequences related to rings.

From Bernard Schott, Mar 28 2021: (Start)
   --------------------------------------------------------------------
   |  Rings with    |  with 1   | without 1 |       with 1 or         |
   |  n elements    |           |           |       without 1         |
   --------------------------------------------------------------------
   |  Commutative   |  A127707  |  A342375  |        A037289          |
   --------------------------------------------------------------------
   | Noncommutative |  A127708  |  A342376  |        A209401          |
   --------------------------------------------------------------------
   | Commutative or |  A037291  |  A342377  | this sequence: a(0) = 1 |
   | noncommutative |           |           |   A037234 with a(0) = 0 |
   --------------------------------------------------------------------
(End)

apply( A027623(n, e=0)=if( !e, vecprod([call(self(), f) | f <- factor(n)~]), e<3, [2^(n>0), 11][e], e==3, if(n>2, 3*sqrtnint(n, 3), 2)+50, n>2 || e>4, /*error*/("not yet implemented"), 390), [0..63]) \\ M. F. Hasler, Jan 05 2021

More terms from Christian G. Bower, Jun 15 1998

a(16) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000

A037289 Number of commutative rings with n elements.

Original entry on oeis.org

1, 2, 2, 9, 2, 4, 2, 34, 9, 4, 2, 18, 2, 4, 4, 162, 2, 18, 2, 18, 4, 4, 2, 68, 9, 4, 36, 18, 2, 8, 2

Offset: 1

Christian G. Bower, Jun 15 1998

These rings do not necessarily contain an identity element.

This sequence is multiplicative. See the reference "The Numbers of Small Rings" below, which proves the result for all rings; restricting to commutative rings only makes the proof easier. - Conjecture by Mitch Harris, Apr 19 2005, proof found by Franklin T. Adams-Watters, Jul 10 2012

Simon R. Blackburn and K. Robin McLean, The enumeration of finite rings, 2022 preprint. arXiv:2107.13215 [math.CO]
A. V. Lelechenko, Parity of the number of primes in a given interval and algorithms of the sublinear summation, arXiv preprint arXiv:1305.1639, 2013
C. Noebauer, Home page [Archived copy as of 2008 from web.archive.org]
Christof Noebauer, The numbers of small rings (PostScript).
C. Noebauer, Thesis on the enumeration of near-rings
Bjorn Poonen, The moduli space of commutative algebras of finite rank, J. Eur. Math. Soc. (JEMS) 10:3 (2008), pp. 817-836. arXiv:0608491 [This contains an error, see Blackburn & McLean]

Cf. A027623, A037291.

a(p^n) = p^(2/27 * n^3 + O(n^2.5)), see Blackburn & McLean. - Charles R Greathouse IV, Jul 13 2022

a(16) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000, who reports that the sequence continues a(32) = ? (> 876), a(33) = 4, 4, 4, 81, 2, 4, 4, 68, 2, 8, 2, 18, 18, 4, 2, 324, 9, 18, 4, 18, 2, 72, 4, 68, 4, 4, 2, 36, 2, 4, 18 = a(63), a(64) = ? (> 12696)

A127707 Number of commutative rings with 1 containing n elements.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 10, 4, 1, 1, 4, 1, 1, 1, 37, 1, 4, 1, 4, 1, 1, 1, 10, 4, 1, 11, 4, 1, 1, 1, 109, 1, 1, 1, 16, 1, 1, 1, 10, 1, 1, 1, 4, 4, 1, 1, 37, 4, 4, 1, 4, 1, 11, 1, 10, 1, 1, 1, 4, 1, 1, 4

Offset: 1

Hugues Randriam (randriam(AT)enst.fr), Jan 24 2007

Is this a multiplicative function?

Answer: yes! See the Eric M. Rains link for a proof for the result for all unital rings; restricting to commutative rings does not affect the essence of the proof. - Jianing Song, Feb 02 2020

C. Noebauer, Home page and Table of numbers of small rings [Archived copies from web.archive.org]
Eric M. Rains, The number of unital rings with n elements is a multiplicative function of n.

Cf. A037289, A037291, A127708, A027623, A307000, A341202.

a(n) = A037291(n) - A127708(n). - Bernard Schott, Apr 19 2022

Keyword 'mult' added by Jianing Song, Feb 02 2020
a(32)-a(63) using Nöbauer's data added by Andrey Zabolotskiy, Apr 18 2022
a(32) = 109 corrected by Bernard Schott, Apr 19 2022

A127708 Number of non-commutative rings with 1 containing n elements.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 99, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Hugues Randriam (randriam(AT)enst.fr), Jan 24 2007, Jan 29 2007

We consider rings in which multiplication is associative and has a unit, but where there is at least one pair of non-commuting elements.

			a(n)=0 for n<=7 and a(8)=1, so all rings (with unit) of cardinality at most 7 are commutative, while the smallest non-commutative ring (with unit) has cardinality 8 and is unique up to isomorphism; it can be represented as the ring of upper-triangular matrices of size 2 over F_2.
A037291(32) = 208, A127707(32) = 109, hence a(32) = 208 - 109 = 99.

C. Noebauer, Home page [Archived copy as of 2008 from web.archive.org]

Cf. A027623, A037291, A127707, A209401, A342376.

a(n) = A037291(n) - A127707(n). - Bernard Schott, Apr 19 2022

a(32)-a(63) from Bernard Schott, Apr 19 2022

a(54) corrected by Andrey Zabolotskiy, Feb 02 2023

A339948 Number of non-isomorphic generalized quaternion rings over Z/nZ.

Original entry on oeis.org

1, 1, 4, 7, 4, 16, 4, 16, 10, 16, 4, 40, 4, 16, 16, 36, 4, 40, 4, 40, 16, 16, 4, 80, 10, 16, 20, 40, 4, 64, 4, 52, 16, 16, 16

Offset: 1

José María Grau Ribas, Dec 24 2020

Generalized quaternion rings over Z/nZ are of the form Z_n/(x^2-a, y^2-b, xy+yx).

			For n=2 all such rings are isomorphic to Z_n<x,y>/(x^2, y^2, xy+yx), so a(2)=1.
For n=4:
  Z_n<x,y>/(x^2,   y^2,   xy+yx),
  Z_n<x,y>/(x^2,   y^2-1, xy+yx),
  Z_n<x,y>/(x^2,   y^2-2, xy+yx),
  Z_n<x,y>/(x^2,   y^2-3, xy+yx),
  Z_n<x,y>/(x^2-1, y^2-1, xy+yx),
  Z_n<x,y>/(x^2-1, y^2-2, xy+yx),
  Z_n<x,y>/(x^2-3, y^2-3, xy+yx),
so a(4)=7.

José María Grau Ribas, Generalized quaternion rings over Z/nZ for an odd n
Jose María Grau, C. Miguel and A. M. Oller-Marcen, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017.
J. M. Grau, C. Miguel and A. M. Oller-Marcén, Generalized quaternion rings over Z/nZ for an odd n, Advances in Applied Clifford Algebras, 28(1), (2018) article 17.

Cf. A286779, A027623, A037221, A127708, A037291, A127707, A038538, A037289, A037290, A038036.

Clear[phi]; phi[1] = phi[2] = 1; phi[4] = 7; phi[8] = 16;
phi[16] = 36; phi[p_, s_] := 2 s^2 + 2;
phi[n_] :=  Module[{aux = FactorInteger[n]},Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];
Table[phi[i], {i,1, 35}]

If n is odd then a(n) = A286779(n).

A341201 Number of unitary rings with additive group (Z/nZ)^3.

Original entry on oeis.org

1, 7, 7, 27, 7, 49, 7

Offset: 1

José María Grau Ribas, Feb 07 2021

R. Gilmer and J. Mott, Associative rings of order_P^3, Proc. Jpn. Acad., 49, No. 10, (1973), 725-799.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

Cf. A341202, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948.

A341202 Number of unitary commutative rings with additive group (Z/nZ)^3.

Original entry on oeis.org

1, 6, 6, 16, 6, 36, 6

Offset: 1

José María Grau Ribas, Feb 07 2021

R. Gilmer and J. Mott, Associative rings of order P^3, Proc. Jpn. Acad., 49, No. 10, (1973), 725-799.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

Cf. A341201, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948.

A209401 Number of noncommutative rings with n elements.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 18, 2, 0, 0, 4, 0, 0, 0, 228, 0, 4, 0, 4, 0, 0, 0, 36, 2, 0, 23, 4, 0, 0, 0

Offset: 1

Ben Branman, Mar 26 2012

a(n)=0 if and only if n is squarefree.

			For n=8, there are 52 rings of order 8, 18 of which are noncommutative, so a(8)=18.

Gregory Dresden, Rings of Size Four, demonstrating that a(4)=2.

Cf. A027623, A037289, A037291.

a(n) = A027623(n) - A037289(n).

A342376 Number of non-commutative rings without 1 containing n elements.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 17, 2, 0, 0, 4, 0, 0, 0, 215, 0, 4, 0, 4, 0, 0, 0, 35, 2, 0, 23, 4, 0, 0, 0

Offset: 1

Bernard Schott, Mar 10 2021

A ring without 1 is still a ring, although sometimes called a rng, or a non-unital ring, or a pseudo-ring (see Wikipedia links).
These are rings in which multiplication has no unit, and where there is at least one pair of non-commuting elements.
a(n)=0 if and only if n is squarefree.

			For n=4, there are 11 rings of order 4, 2 of which are without 1 and non-commutative, so a(4)= 2. Note that for these 2 rings, the abelian group under addition is the commutative Klein group Z/2Z + Z/2Z. These two rings are the last two rings described in the link _Greg Dresden_ in reference: Ring 22.NC.1 and Ring 22.NC.2.

Gregory Dresden, Rings of Size Four.
Wikipedia, Pseudo-ring.
Wikipedia, Rng.
Index to sequences related to rings.

Number of non-commutative rings: A127708 (with 1 containing n elements), this sequence (without 1 containing n elements), A209401 (with n elements).
Cf. A127707, A342375, A037289, A037291, A342377, A027623, A037234.
Cf. A005117, A013929.

a(n) = A209401(n) - A127708(n) = A342377(n) - A342375(n).

a(A005117(n)) = 0; a(A013929(n)) > 0.

a(28) corrected by Des MacHale, Mar 20 2021

A037221 Number of near-rings (or nearrings) definable on cyclic group of order n.

Original entry on oeis.org

3, 5, 12, 10, 60, 24, 135, 222, 329, 139, 1749, 454, 2716, 3817

Offset: 2

N. J. A. Sloane

W. E. Clark and J. Pedersen, Catalogue of Algebraic Systems
C. Noebauer, Home page [Archived copy as of 2008 from web.archive.org]
C. Noebauer, The Numbers of Small Rings
C. Noebauer, Thesis on the enumeration of near-rings

Cf. A027623, A037291.

Corrected by Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000

cp's OEIS Frontend

A027623 a(0) = 1; for n > 0, a(n) = number of rings with n elements.

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A037289 Number of commutative rings with n elements.

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A127707 Number of commutative rings with 1 containing n elements.

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A127708 Number of non-commutative rings with 1 containing n elements.

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A339948 Number of non-isomorphic generalized quaternion rings over Z/nZ.

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A341201 Number of unitary rings with additive group (Z/nZ)^3.

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A341202 Number of unitary commutative rings with additive group (Z/nZ)^3.

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A209401 Number of noncommutative rings with n elements.

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A342376 Number of non-commutative rings without 1 containing n elements.

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