A lot of books/professors start with rings instead of groups because a lot of the proofs or properties of groups involve number theory, primes, irreducibles, euclidean algorithm, division algorithm, etc which follow from ring theory.
Specifically Dummit and Foote define a ring as follows:
A ring $R$ is a set together with two binary operations $+$ and $\times$ (called addition and multiplication) (which just means the operations are closed, so if $a, b \in R$, then $a + b \in R$ and $a \times b \in R$) satisfying the following axioms:
- $(R, +)$ is an abelian group.
- $\times$ is associative: $(a \times b) \times c = a \times(b \times c)$ for all $a, b, c \in R$.
- The distributive law holds in $R$: for all $a, b, c\in R$ $$(a + b) \times c = (a \times c) + (b \times c) \text{ and } a \times (b + c) = (a \times b) + (a \times c)$$
By saying $(R, +)$ is an abelian group, we mean there is an identity element $0$ such that $a + 0 = a$ for all $a \in R$, there is an additive inverse $-a$ such that $a + (-a) = 0$ for all $a \in R$ and addition is associative and commutative.
