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

Author

Bernard Schott, Mar 10 2021

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

• Gregory Dresden, Rings of Size Four.
• Wikipedia, Pseudo-ring.
• Wikipedia, Rng.
• Index to sequences related to rings.

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.

Formula

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

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

Extensions

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