Let $m$ be an order function, and let $S(m)$ be the corresponding semiclassical symbol class. Assume $S(m)$ is viewed as a vector space and that $hS(m) := \{h b : b \in S(m)\}$ is a linear subspace of $S(m)$. Let $\iota: hS(m) \to S(m)$ be the inclusion map defined by $\iota(c)=c$, and let $\pi: S(m) \to S(m)/hS(m)$ be the quotient map defined by $\pi(a)=a+hS(m)$. Then the sequence $0 \longrightarrow hS(m) \xrightarrow{\iota} S(m) \xrightarrow{\pi} S(m)/hS(m) \longrightarrow 0$ is exact. Consequently, if the principal symbol map is the quotient map $\sigma_{\mathrm{pr}}: S(m) \to S(m)/hS(m)$ defined by $\sigma_{\mathrm{pr}}(a) := a+hS(m)$, then $\sigma_{\mathrm{pr}}(a)=0$ if and only if $a \in hS(m)$.