A ring $(R, \cdot, * )$ is a set $R$ equipped with two binary operations (say $\cdot$ and $*$) which satisfy the following properties:
$R$ forms an abelian group under $\cdot$, i.e.,
(i) $R$ is closed under $\cdot$ $(\textbf{A}_1)$
(ii) $\cdot$ is associative in $R$ ($\textbf{A}_2$)
(iii) $\cdot$ has an identity in $R$ ($\textbf{A}_3$)
(iv) all $r \in R$ have inverses $r^{-1} \in R$ under $\cdot$ ($\textbf{A}_4$)
(v) $\cdot$ is commutative in $R$ ($\textbf{A}_5$)
$R$ forms a semigroup under $$, *i.e.,
(i) $R$ is closed under $*$ ($\textbf{M}_1$)
(ii) $R$ is associative in $R$ ($\textbf{M}_2$)
$* $ distributes over $\cdot$ in $R$ ($\textbf{M}_3$), i.e.,
(i) $a * (b \cdot c) = (a * b) \cdot (a * c)$,
(ii) $(a \cdot b ) * c = (a * c) \cdot (b * c)$,
$\forall a,b,c \in R$.
$(R, \cdot, * )$ is called commutative if $*$ is commutative in $R$. ($\textbf{M}_4$)
$(R, \cdot, * )$ is said to be a ring with identity if $*$ has an identity in $R$.
The set of even integers under addition and multiplication, as well as the set of $n \times n$ matrices under matrix addition and multiplication, are examples of fields.
A field $(F, +, \times)$ is a set $F$ equipped with two binary operations $+$ and $\times$ which satisfy the following properties:
$F$ is an integral domain, i.e.,
(i) $F$ forms a commutative ring under $+$ and $\times$, i.e., ($\textbf{A}_1-\textbf{A}_4, \textbf{M}_1-\textbf{M}_4$)
(ii) $\times$ has an identity in $F$ (the multiplicative identity), i.e., $\exists 1 \in F$ such that $a \times 1 = 1 \times a = a$, $\forall a \in F$. ($\textbf{M}_5$)
(iii) $F$ has no zero divisors, i.e., $ab = 0 \Rightarrow (a = 0) \lor (b = 0)$, $\forall a,b \in F$. ($\textbf{M}_6$)
all $f \in F$ (except 0) have multiplicative inverses $f^{-1} \in F$ such that $ff^{-1} = f^{-1}f = 1$. ($\textbf{M}_7$)
The sets of real numbers, rational numbers and complex numbers all form fields under addition and multiplication.
$Z_n$ is a commutative ring with identity (under addition and multiplication mod $n$), but not a field unless $n$ is prime (if $n$ has non-trivial factors $a, b$, then $ab = 0$ but $a \neq 0$ and $b \neq 0$, so there are zero divisors).
For prime $p$, the field $(Z_p, +_p, \cdot_p)$ is called a Galois field and also denoted as $GF(p)$.