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

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• nice
• hard
• more
• mult

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

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)

Programs

PARI

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

Extensions

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