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For integer $n \geq 2$, the number of rngs of size $n$ is in general an open problem.

Let $a(n)$ be the number of commutative rngs of size $n$. We can use the facts that $a(n)$ is multiplicative and that $a(p^n) = p^{2/27 * n^3 + O(n^{8/3})}$, for prime $p$, to estimate $a(n)$ for general $n$. These rings do not necessarily have identity. https://oeis.org/A037289

Let $b(n)$ be the number of rings with identity of size $n$. These rings are not necessarily commutative. A few terms of this sequence are known: https://oeis.org/A037291

Let $c(n)$ be the number of commutative rings with identity. We know commutative rings with identity are direct products of local rings (rings with a unique maximal ideal), so if we knew the number of local rings of a given (prime-power) size, we could find $c(n)$.

We know $c(n) \leq a(n)$ and $c(n) \leq b(n)$, and some crude lower bounds on $c(n)$ can be found by considering classes of rings of size $n$, such as direct sums of integer rings.

I know all of the local rings of size $p^n$ are known for $n=1,2,3,4,5$, but are there bounds on the number of commutative rings of size $n$ or the number of local rings of size $p^n$ for general $n$?

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