Suppose that $F$ is a field and $R=F⊕x^2F[[x]]$, where $F[[x]]$ is the ring of power series in one indeterminate $x$ with coefficients in $F$.
I guess that $R$ is a local ring with the maximal ideal $(x^2,x^3)$, and have three issues in this regard: Is my guess true? Is this a minimal generating set? Are there other minimal generating sets?
Thanks very much for anybody concerning this problem!
Your guess is correct. You should be able to show that $R/(x^2,x^3) = F$, which implies that $(x^2,x^3)$ is indeed maximal.
To show that this is the only maximal ideal, it is necessary and sufficient to show that every element outside of $(x^2,x^3)$ is a unit. You can do this by noting that the procedure when working in $F[[x]]$ works here, i.e. the inverse you end up with is indeed in $R$.
If $(x^2,x^3)$ is not a minimal generating set, then the ideal is principal, say given by $(f)$. What is the degree of the lowest term of $f$? Reach a contradiction from here.
Other generating sets are given by $(\varepsilon x^2, \eta x^3)$, where $\varepsilon$ and $\eta$ are units in $R$. To see that these are the only other generating sets, write $(x^2,x^3) = (a,b)$ so that we must have $r,s \in R$ with $x^2 = ra + sb$. Note (similar to before) that the lowest degree of $a$ and $b$ must be different (one is 2, the other 3), so that either $r = 0$ or $s = 0$. The statement quickly follows.