Coprimeness over $mathcal{RH}_{infty}$ (a set of stable and proper rational functions) is an important concept for stabilizing controller synthesis. However, in my view, coprimeness is also hard for postgraduate students majoring in control science to understand. Here, I would like to show the coprimeness over different sets (principal ideal domains) in a unified framework, respectively.
When we were in primary school or high school, we were told that two integer are coprime if their greastest common divisor (g.c.d.) is $1$. Equivalently, (as the set of integers $mathbb{Z}$ is a principal ideal domain) $a$ and $b$ are coprime if and only if there exist two integers $c$ and $d$ such that
egin{equation*}
a c + bd = 1.
end{equation*} For example, $2$ and $3$ are coprime since $2 imes (-1) + 3 imes 1 = 1$ or $2 imes (-4) + 3 imes 3 = 1$. This implies that $c$ and $d$ are not unique.
Note that $1$ (or $-1$) is also called the unit in $mathbb{Z}$, as $1$ and $-1$ are the only two integers have inverse in $mathbb{Z}$. Thus, we can say that two integers are coprime if their g.c.d. is a unit. This allows us to generalize the coprimeness to polynomials and rational functions.
It is well-known that two (real) polynomials $p(s)$ and $q(s)$ are coprime if they have no common roots, i.e., if their g.c.d. is a zero-order polynomial (or, nonzero constant $mathbb{R}ackslash {0}$). It is worth to figure out that the nonzero constants are the only invertible polynomials so that they are units of polynomials.
For example, the real polynomials $s$ and $1$ are coprime as $0 imes s + 1 imes 1 = 1$ or $x imes s + (-xs+1) imes 1 = 1$ for any $x in mathbb{R}$.
Similarly, for polynomial matrices, the so-called unimodular matrices are the only polynomial matrices invertible in the set of polynomial matrices. Therefore, the two polynomial matrices are coprime if their g.c.d. is a unimodular matrix.