A127707 -id:A127707 - 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

A037291 Number of rings with 1 containing n elements.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 11, 4, 1, 1, 4, 1, 1, 1, 50, 1, 4, 1, 4, 1, 1, 1, 11, 4, 1, 12, 4, 1, 1, 1, 208, 1, 1, 1, 16, 1, 1, 1, 11, 1, 1, 1, 4, 4, 1, 1, 50, 4, 4, 1, 4, 1, 12, 1, 11, 1, 1, 1, 4, 1, 1, 4

Offset: 1

Christian G. Bower, Jun 15 1998

Many authors simply call these "rings". They are also known as unital rings, rings with unity, or rings with identity. - Charles R Greathouse IV, Aug 12 2015

Is this sequence multiplicative? That is, if p and q are distinct primes, is it true that a(p^i*q^j) = a(p^i)*a(q^j)? - Jianing Song, Oct 26 2019. The answer is yes - see the Eric M. Rains link. - N. J. A. Sloane, Oct 27 2019

Gérard P. Michon, Ring Theory, Numericana, 2000-2022.
C. Noebauer, Home page and Table of numbers of small rings [Archived copies]
Eric M. Rains, The number of unital rings with n elements is a multiplicative function of n.

Cf. A027623, A037221, A127707.

a(16) and a(32)-a(63) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000
Keyword 'mult' added by Jianing Song, Feb 02 2020
a(54) corrected by Andrey Zabolotskiy, Feb 02 2023

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.

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

A341547 Number of rings with additive group (Z/nZ)^2.

Original entry on oeis.org

1, 8, 8, 66, 8, 64, 8, 301, 175, 64, 8, 528, 8, 64, 64

Offset: 1

José María Grau Ribas, Feb 14 2021

B. Fine, Classification of Finite Rings of Order p^2, Mathematics Magazine, 66(4) (1993), 248-252.
C. R. Fletcher, Rings of Small Order, The Mathematical Gazette, 64(427) (1980), 9-22.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

Cf. A341548, A341201, A341202, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948, A000005.

Clear[phi]; phi[1] = 1; phi[p_,1] := 8; phi[2,2] = 66;
phi[2,3] = 301; phi[3,2] = 175; phi[n_]:= Module[{aux = FactorInteger[n]}, Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];

A341548 Number of commutative rings with additive group (Z/nZ)^2.

Original entry on oeis.org

1, 6, 6, 28, 6, 36, 6, 79, 35, 36, 6, 168, 6, 36, 36

Offset: 1

José María Grau Ribas, Feb 14 2021

It appears that a(16)=230, but it is preferable to wait for someone to confirm it.

B. Fine, Classification of Finite Rings of Order p^2, Mathematics Magazine, 66(4) (1993), 248-252.
C. R. Fletcher, Rings of Small Order, The Mathematical Gazette, 64(427) (1980), 9-22.
Christof Nöbauer, Numbers of rings on groups of prime power order
Christof Nöbauer, Home page

Cf. A341547, A341201, A341202, A037234, A037291, A127708, A127707, A037289, A037290, A038036, A307000, A339948, A000005.

Clear[phi]; phi[p_, 1] := 6;  phi[2,2] = 28; phi[2,3] = 79;  phi[3,2] = 35; phi[n_]:= Module[{aux = FactorInteger[n]}, Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];

A342375 Number of commutative rings without 1 containing n elements.

Original entry on oeis.org

0, 1, 1, 5, 1, 3, 1, 24, 5, 3, 1, 14, 1, 3, 3, 125, 1, 14, 1, 14, 3, 3, 1, 58, 5, 3, 25, 14, 1, 7, 1

Offset: 1

Bernard Schott, Mar 09 2021

A ring without 1 is still a ring, but sometimes it is called a rng, or a non-unital ring, or a pseudo-ring (see Wikipedia links).

			a(1) = 0 because the only ring with 1 element is the zero ring with the element 0, and for this ring, 0 and 1 coincide.
a(2) = 1, and for this corresponding ring with elements {0,a}, the multiplication that is defined by: 0*0 = 0*a = a*0 = a*a = 0 is commutative, also this ring is without unit, hence a(2) = 1; the Matrix ring {0,a} with coefficients from Z/2Z:
          (0 0)           (0 0)
      0 = (0 0)       a = (1 0)  is such an example.
For n=8, there are 52 rings of order 8, 24 of which are commutative rings without 1, so a(8) = 24.

Number of commutative rings: A127707 (with 1 containing n elements), this sequence (without 1 containing n elements), A037289 (with n elements).

Cf. A127708, A342376, A209401, A037291, A342377, A027623, A037234.

a(n) = A037289(n) - A127707(n).

cp's OEIS Frontend

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

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A037291 Number of 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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A342376 Number of non-commutative rings without 1 containing n elements.

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A341547 Number of rings with additive group (Z/nZ)^2.

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A341548 Number of commutative rings with additive group (Z/nZ)^2.

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