A342375 - OEIS

%I #22 Mar 11 2021 20:59:59

%S 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,

%T 1

%N Number of commutative rings without 1 containing n elements.

%C 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).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pseudo-ring">Pseudo-ring</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rng_(algebra)">Rng</a>.

%H <a href="/index/Res#rings">Index to sequences related to rings</a>.

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

%e 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.

%e 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:

%e           (0 0)           (0 0)

%e       0 = (0 0)       a = (1 0)  is such an example.

%e For n=8, there are 52 rings of order 8, 24 of which are commutative rings without 1, so a(8) = 24.

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

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

%K nonn,more

%O 1,4

%A _Bernard Schott_, Mar 09 2021