I have a fair number of books on ring theory, and the term "ring" universally means a ring that may or may not have a 1. (That is also the convention in the OEIS.) No need to invent a Hebrew-style vowel-less name! The books by T Y Lam are the classics. There was a time when codes over rings (rather than fields) were all the rage. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Tue, Jan 5, 2021 at 9:23 PM Allan Wechsler <acwacw@gmail.com> wrote:
Oh, regarding Marc LeBrun's interesting speculations on the number of rings (or rngs) that can be built with a given additive structure: remember that addition is required to be commutative in rngs.
Are there any abelian groups that support no rngs?
On Tue, Jan 5, 2021 at 9:18 PM Allan Wechsler <acwacw@gmail.com> wrote:
It's easy to get tripped up by the terms of art. Apparently there is a word for a structure that is like a ring but is not required to have a multiplicative identity: a rng (pronounced "rung"). But that hasn't stopped some mathematicians from using "ring" for this concept, saying "ring with unit" when they mean "ring".
Alas, this means that *every* time OEIS talks about rings, we have to clarify whether we mean to include rngs. A027623 is quite explicit and exemplary.
On Tue, Jan 5, 2021 at 9:06 PM Neil Sloane <njasloane@gmail.com> wrote:
BTW, he isn't assuming the ring has a unit, and IIRC he says that much less is known about that case.
In case someone is going to program this.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Jan 5, 2021 at 8:49 PM Marc LeBrun <mlb@well.com> wrote:
there are infinitely many n such that there are more groups with n elements than rings with n elements
Hm, I'm woefully uneducated in this stuff, but for this to even be an interesting question I think this says there must be groups that can be extended to rings in more than one way (since every ring an additive group augmented with a multiplicative operator).
That suggests a table for the OEIS: T[n,k] = number of groups of order n that can be extended to (at most) k different rings.
T[n,0] would be the number of groups of order n that cannot be augmented to be rings.
Row sum over k of T[n,k] = number of groups of order n.
Row sum over k of k * T[n,k] = number of rings of order n
I'm curious: what is the smallest group G that supports more than one ring? (ie a G with min n such that T[n, some k > 1] is > 0)
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