A342376 Number of non-commutative rings without 1 containing n elements.

0, 0, 0, 2, 0, 0, 0, 17, 2, 0, 0, 4, 0, 0, 0, 215, 0, 4, 0, 4, 0, 0, 0, 35, 2, 0, 23, 4, 0, 0, 0

Offset

1,4

Comments

A ring without 1 is still a ring, although sometimes called a rng, or a non-unital ring, or a pseudo-ring (see Wikipedia links).

These are rings in which multiplication has no unit, and where there is at least one pair of non-commuting elements.

a(n)=0 if and only if n is squarefree.

Links

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

Gregory Dresden, Rings of Size Four.

Wikipedia, Pseudo-ring.

Wikipedia, Rng.

Index to sequences related to rings.

Formula

a(n) = A209401(n) - A127708(n) = A342377(n) - A342375(n).

a(A005117(n)) = 0; a(A013929(n)) > 0.

Example

For n=4, there are 11 rings of order 4, 2 of which are without 1 and non-commutative, so a(4)= 2. Note that for these 2 rings, the abelian group under addition is the commutative Klein group Z/2Z + Z/2Z. These two rings are the last two rings described in the link Greg Dresden in reference: Ring 22.NC.1 and Ring 22.NC.2.

Crossrefs

Number of non-commutative rings: A127708 (with 1 containing n elements), this sequence (without 1 containing n elements), A209401 (with n elements).

Cf. A127707, A342375, A037289, A037291, A342377, A027623, A037234.

Cf. A005117, A013929.

Sequence in context: A224074 A136615 A283950 * A277443 A209401 A029696

Adjacent sequences: A342373 A342374 A342375 * A342377 A342378 A342379

Keyword

nonn,more

Author

Bernard Schott, Mar 10 2021

Extensions

a(28) corrected by Des MacHale, Mar 20 2021

Status

approved