Let us fix a field $K$, and let us consider the principal ideal ring $K[x]$. If I is an ideal, since $K[x]$ is a PID, we can write $I = (p(x))$ for some polynomial $p(x)$. Now, let us say that a set of generators $S= \{x_1, \ldots , x_n\}$ for I is irredundant if no subset of S of cardinality less than n generates I.
Can we construct, for every $n \geq 1$, and any non-zero ideal an irredundant set of generators of cardinality $n$? For example, for $n= 2$, the ideal $(x)$ has an irredundant set of generators $\{x+x^2,x^2\}$. I have tried doing some induction, but it didn't amount to much, and neither did explicitly trying to construct an irredundant set for an arbitrary non-zero ideal. So, does anyone see how to do it, and if so, any hints?