In $K[x]$ (where $K$ is a field), I know that every ideal can be written as $(f)$ for some $f \in K[x]$. Furthermore, $f$ is unique up to multiplication by a nonzero constant in $K$.
Is there a analogous uniqueness property for generator of principal ideal in $K[x_1,x_2,...,x_n]$? If the answer is no, could anyone provide me a counterexample?
On
To complement the answer by MooS, let $ R $ be a domain and let $ (x) = (y) $ as principal ideals. Then, $ x = k_1 y $ and $ y = k_2 x $ for some $ k_1, k_2 \in R $, and hence $ x = k_1 y = k_1 k_2 x $. Since $ R $ is a domain, it follows that $ k_1 k_2 = 1 $, hence both $ k_1 $ and $ k_2 $ are units in $ R $. The result now follows.
In any domain, the generator of a principal ideal is unique upto multiplication with units. Note that - for any $n$ - the units of $K[x_1, \dotsc, x_n]$ are the nonzero constants in $K$. Thus we have exactly the same property (With the difference being that not any ideal is principal if $n \geq 2$).