how to prove a ring is subring?

example:

how to prove 3Z is subring of Z (Z is integer)

we only need to prove 3Z is closed under subtraction and multiplication.

subtraction => 0 and additive inverse in the 3Z.

let a, b in 3Z.

if a – b in 3Z then

let a = b, =>  b + (-b)  = in 3Z

=> 0 in 3Z and additive inverse -b in 3Z