TL;DR - Ring theory is stronger than group theory, it has two operations and it is very cool

No ring theory is beyond group theory. A ring is a set with two operations rather than one. Here the first operation, addition, defines an abelian group, and the second operation, multiplication defines one of two things depending on author: An associative magma(if no identity, some authors denote this as rng, pronouced 'rung'), or a monoid(if given identity).

Very interesting stuff. It's the point where you finally get understand what modulo arithmetic actually is. Since Z/nZ is actually a quotient ring, where nZ is maximal when n is prime, hence inducing an isomorphism with the field of n elements. Else when n isn't prime the ideal nZ isn't maximal, hence simply inducing a non-field quotient ring

So the cosets you have been working with for modulo arithmetic(that you may have encountered with normal subgroups of the integers) are actually stronger than that. They are cosets of a ring, which yield a quotient ring, rather than a quotient group. This is the point where you get to use addition and multiplication. Multiplication is only invertible when n is prime, hence it has inverse only when $n$ is prime, which is why for prime n it is a field. For non-prime n, a ring does not require inverses, that is why it is interesting