A342375 Number of commutative rings without 1 containing n elements.

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

1,4

Comments

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).

Links

Table of n, a(n) for n=1..31.

Wikipedia, Pseudo-ring.

Wikipedia, Rng.

Index to sequences related to rings.

Formula

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

Example

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.

Crossrefs

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.

Sequence in context: A206076 A329374 A115638 * A055515 A363437 A338096

Adjacent sequences: A342372 A342373 A342374 * A342376 A342377 A342378

Keyword

nonn,more

Author

Bernard Schott, Mar 09 2021

Status

approved