greatest common divisor for polynomials

Question

Let $k$ be a field and $f(x), g(x)\in k [x]$. The greatest common divisor is defined to the monic common divisor having largest degree. In this definition, can the condition $k$ being a field be weakened to $k$ being a commutative ring? That is, is it well-defined due to the uniqueness?

Answer 1

For example, the gcd of $2x+1$ and $(2x+1)x$ in $\mathbb Q[x]$ is $x+\frac{1}{2}$, which is not in $\mathbb Z[x]$.

So we cannot just replace $k$ with any commutative ring. This is really because of the monic condition on the gcd; we can divide by the leading entry when working in $k$, but this is not always possible in a general commutative ring.