I dealt with some of the ring sequences in Mitch's first batch, but the problem keeps coming back.

Ok, in

http://www.algebra.uni-linz.ac.at/~noebsi/pub/smallrings.ps

I find

Theorem 1. [Sho38] Every finite ring R can be uniquely (up to isomorphism) decomposed into a direct sum of rings of prime power order.

The Noebauer paper should settle that A027623 and A037234 thus, the ring sequence (as well as commutative, with unit, etc) is multiplicative. This covers A027623 and A037234. I strongly suspect this is also true of A037292 (Non associative rings) but I haven't had time to do the algebra myself to confirm it and obviously the theorem does not address it. Regarding Dave's observation:

In A027623 (number of rings with n elements), I noticed that for p prime, a(p) = 2 and a(p^2) = 11 seem to be true. However, for p^3, we have a(8) = 52 while a(27) = 59, so a(p^3) depends on p. Then I found this enticing tidbit:

The paper by Antipkin/Elizarov also gives the number a(p^3) of rings of order p^3. - Hans H. Storrer (storrer(AT)math.unizh.ch), Sep 16 2003 ... If A027623(p^3) were in the OEIS, it would start 52,59,... But no such sequence exists in the OEIS.

The Noebauer paper also states that

THEOREM 4. [FW74,LW80] There are (3p+4)n-4p+11 non-isomorphic rings with additive group Z/p^n + Z/p (n>=2) if p is odd and 9n+2 if p=2.

THEOREM 5. [FW74,LW80] There are p+27 non-isomorphic rings with additive group Z/p + Z/p + Z/p if p is odd and 28 if p=2.

These two theorems give a total of 3p+50 non-isomorphic rings of order p^3 for odd p and 52 for p=2. This result is also obtained by [AE83].

That result is given in the A027623 entry. Christian