= Solution
The commutative <Laurent polynomial ring> $B$ is an <integral domain> and $\sigma$ is invertible. Write nonzero elements as
$$
r=\sum_{i=m_0}^{m}a_ix^i,\qquad s=\sum_{j=n_0}^{n}b_jx^j,\qquad a_m,b_n\ne0.
$$
The largest exponent in their product is $m+n$, with coefficient
$$
\boxed{a_m\sigma^m(b_n)\ne0}.
$$
There is only one pair of exponents contributing at this degree, and the coefficient cannot vanish in the <integral domain> $B$. Hence $rs\ne0$. Equivalently, \b[$rs=0$ forces $r=0$ or $s=0$.] This is <skew Laurent extensions of domains are domains>, and the argument works over the original <field> in any characteristic.
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