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

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%I A127708 #23 Feb 02 2023 15:29:23
%S A127708 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,
%T A127708 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
%N A127708 Number of non-commutative rings with 1 containing n elements.
%C A127708 We consider rings in which multiplication is associative and has a unit, but where there is at least one pair of non-commuting elements.
%H A127708 C. Noebauer, <a href="https://web.archive.org/web/20080111141811/http://www.algebra.uni-linz.ac.at/~noebsi/">Home page</a> [Archived copy as of 2008 from web.archive.org]
%F A127708 a(n) = A037291(n) - A127707(n). - _Bernard Schott_, Apr 19 2022
%e A127708 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.
%e A127708 A037291(32) = 208, A127707(32) = 109, hence a(32) = 208 - 109 = 99.
%Y A127708 Cf. A027623, A037291, A127707, A209401, A342376.
%K A127708 nonn,nice,more
%O A127708 1,16
%A A127708 Hugues Randriam (randriam(AT)enst.fr), Jan 24 2007, Jan 29 2007
%E A127708 a(32)-a(63) from _Bernard Schott_, Apr 19 2022
%E A127708 a(54) corrected by _Andrey Zabolotskiy_, Feb 02 2023