On a set $A$ (not empty), additional operation and
multiplicative operation are defined.
If $a,b\in A$ ,
(1)$\rho_1:(a,b)\rightarrow a\rho_1b\in A$
(2)$\rho_2:(a,b)\rightarrow a\rho_2b\in A$
You may prefer the expression $a+b$ to $a\rho_1b$ and $a\times b$ to $a\rho_2b$ .
In addition, the following conditions are satisfied.
(1)for the additional operation $\rho_1$ , $(A,\rho_1)$ is the abelian group.
(2)the multiplicative operation $\rho_2$ is associative.
$(a\rho_2 b)\rho_2 c=a\rho_2(b\rho_2 c)\qquad (a,b,c\in A)$
(3)the multiplicative operation $\rho_2$ is distributive
for any addtional operations from both sides.
$a\rho_2(b\rho_1 c)=(a\rho_2 b)\rho_1(a\rho_2 c)$ ,
$(b\rho_1c)\rho_2 a=(b\rho_2 a)\rho_1(c\rho_2 a)$ .
(4)there exists an element $e$ such that $e\rho_2 a=a\rho_2 e=a$ for any $a\in A$ .
Then, $(A,\rho_1,\rho_2)$ is called the ring.
The difference between a ring and a group is that a ring has two operations
and a group has one operation.
If the multiplivative operation is commutative $a\rho_2 b=b\rho_2 a$ ,
then $(A,\rho_1,\rho_2)$ is the abelian ring.
The integer $(\mathbb{Z},+,\times)$ is an abelian ring.
Let $A$ be a family of sets (not empty, but has the empty set).
(1)if $A_i\in A$ ,then $A_i^c\in A$ .
(in other words, $A_i\cup A_j\in A$ and $A_i/A_j\in A$ ($A_i,A_j\in A, i\ne j$)
(2)$\cup_{n=1}^{\infty}A_n\in A$ .
This $A$ is called a $\sigma-$ring .
$\sigma-$ring is one of bases of the lebesgue theory.
