Proof. Let $f(X)$ and $g(X)$ be two non-zero polynomials in $R[X]$ and let $a_f$ and $b_g$ be their leading coefficients, respectively. Thus $a_f \neq 0$ , $b_g \neq 0$ , and because $R$ has no zero divisors, $a_fb_g \neq 0$ . But the product $a_fb_g$ is the leading coefficient of $f(X)g(X)$ and so $f(X)g(X)$ cannot be the zero polynomial. Consequently, $R[X]$ has no zero divisors, Q.E.D.

Remark. The theorem may by induction be generalized for the polynomial ring $R[X_1,\,X_2,\,\ldots,\,X_n]$ .