$15 \cdot 39$ If for any two numbers $a, b$ in a number set $A$, their sum $a+b$, difference $a-b$, and product $a b$ are all within the number set $A$, then the number set $A$ is called a number ring. Among the following six number sets:
(1) $Z=\{$ all integers $\}$
(2) $N=\{$ all natural numbers $\}$
(3) $Q=\{$ all rational numbers $\}$
(4) $R=\{$ all real numbers $\}$
(5) $M=\{$ all numbers of the form $n+m \sqrt{2}$, where $n, m$ are integers $\}$
(6) $P=\left\{\right.$ all numbers of the form $\frac{m}{2 n}$, where $n, m$ are natural numbers $\}$ are number rings:
(A) 6.
(B) 5.
(C) 4.
(D) 3.
(E) 1.
(China Partial Provinces and Cities Junior High School Mathematics Correspondence Competition, 1985)
[Solution] For any integers $a, b$, then $a+b, a-b, ab$ are still integers, so $Z$ is a number ring.

For any natural numbers $m, n$, if $n>m$, then $m-nm, q>p$, then $mp-nq<0$. Knowing that $a-b$ does not belong to $P$, $\therefore P$ is not a number ring.
In summary, only $Z, Q, R, M$ are number rings, i.e., 4 number rings. Therefore, the answer is $(C)$.