I've just started learning about rings. Rings are one additive abelian group strung together, through the associative law, with another structured operation.
Couldn't we continue stringing together operations in this manner (a multi-operation associative law)? Would what I'm thinking be encompassed by the ring definition through something I'm missing? If not, is this done and, if so, do useful objects come out of it?
Best Answer
Absolutely! Such things are studied in a few related disciplines encompassing universal algebra and parts of category theory (including what is called "higher-dimensional algebra"). On the categorical side some keywords here include (algebras over an) operad, (algebras over a) monad, and (models of a) Lawvere theory. More concretely there are also coalgebras, bialgebras, Hopf algebras, Frobenius algebras, etc. Ross Street's Quantum Groups: a path to current algebra was written as an introduction to these more exotic types of algebra that people don't generally bother to tell you about as an undergrad. Far from being esoteric generalizations of ordinary algebra, these structures turn up all over mathematics in the most unexpected places.
As a matter of introduction, however, I don't think that what you've just described is a good way to get a sense of what a ring actually is. The ring axioms capture the abstract properties that symmetries of abelian groups satisfy, in the same way that the group axioms capture the abstract properties that symmetries of sets satisfy. The point here being that if $A$ is an abelian group and $f : A \to A$ some endomorphism of it, then there are two ways that one can combine such endomorphisms: one can compose them or add them. This leads to the multiplication and addition operation, respectively, on the ring $\text{End}(A)$ of endomorphisms of $A$. For example if $A$ is the abelian group $\mathbb{Z}$, then $\text{End}(A)$ is the commutative ring $\mathbb{Z}$.
Then there is a "Cayley's theorem" for rings which says that any abstract ring $R$ can be realized as endomorphisms of some abelian group: in fact $R$ is precisely the ring of endomorphisms of $R$ which respect right multiplication, that is, which respect the right $R$-module structure on $R$.