OFFSET
0,3
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
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.
EXAMPLE
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.
PROG
(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
CROSSREFS
From Bernard Schott, Mar 28 2021: (Start)
--------------------------------------------------------------------
| Rings with | with 1 | without 1 | with 1 or |
| n elements | | | without 1 |
--------------------------------------------------------------------
--------------------------------------------------------------------
--------------------------------------------------------------------
| noncommutative | | | A037234 with a(0) = 0 |
--------------------------------------------------------------------
(End)
KEYWORD
nonn,nice,hard,more,mult
AUTHOR
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
STATUS
approved