The number of unital rings with n elements is a multiplicative function of n.

Eric M. Rains
Professor of Mathematics
The Division of Physics, Mathematics and Astronomy
Caltech

October 27 2019

Theorem: The number of unital rings with n elements is a multiplicative function of n.

There is a natural bijective proof.

      First note that any finite abelian group A has a canonical
decomposition as a product of abelian groups A_p of prime power order,
in which A_p is the subgroup of A consisting of those elements of order
a power of p.  (This implies multiplicativity of the number of abelian
groups.)

      A similar decomposition applies to finite rings.
More precisely, one has the following.

Lemma.  Let R be a finite nonunital ring, and for a prime p, let R_p
denote the p-power subgroup of the additive group of R.  Then R_p is a
subring, and there is a ring isomorphism from R to Prod_p R_p.  Finally,
if R is unital, then so are those R_p which are nonzero.

Proof.  By construction, this contains 0 and is closed under addition
and negation, so it remains only to show it is closed under
multiplication, which follows by observing that if p^k x = 0 and p^j y =
0, then p^{k+j} xy = 0.  Similarly, if x in R_p, y in R_q for distinct
primes p,q, then p^k xy = q^m xy = 0, and thus xy = 0.  In particular, the
direct sum decomposition R = \bigoplus_p R_p as an additive group extends
to a product decomposition as a ring.

Now suppose R is unital.  Then the direct sum decomposition gives us a
unique expression

1 = Sum_p e_p

where each e_p in R_p.  For p,j distinct, e_p e_j = 0, and thus

1 = 1^2 = Sum_p e_p^2,

so that e_p^2 = e_p by uniqueness.  In other words, we have a canonical
decomposition of 1 as a sum of orthogonal idempotents, and thus a
canonical product decomposition into unital rings as desired.
QED