MHB Ideals of formal power series ring

[Anonymous2]

I need help understanding the following solution for the given problem.

The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i \in F$ forms a ring $F[[t]]$. Determine the ideals of the ring.

The solution: Let $I$ be an ideal and $p \in I$ such the number $a := \min\{i|a_i \neq 0\}$ is minimal. We claim $I=(t^a).$ First, $p=t^aq$ for some unit $q$, hence $(t^a) \subset I$. Conversely, any $r \in I$ has first nonzero coefficient at degree $\geq n$, hence $t^a=s$ for some $s \in F[[t]]$, and so $r \in (t^n)$.

My questions: Why the claim $I=(t^a)$? Why does $q$ have to be a unit? What does "first nonzero coefficient at degree $\geq n$ mean? And I don't understand the last part of the proof!

 

[Fallen Angel]

Hi,

My questions: Why the claim $I=(t^a)$?

Just because the author knows what is the answer, he claims it and then prove it.

Why does $q$ have to be a unit?

In a formal power series ring, a series is invertible if and only if the constant term is invertible over the base ring, in this case, $F$ being a field implies a series is invertible (unit) if and only if its constant term is different from zero.

What does "first nonzero coefficient at degree $\geq n$ mean?

This is the same that saying $r=\displaystyle\sum_{i=n}^{\infty}a_{i}t^{i}$, but in fact $n$ should be an $a$, the next sentence make no sense, from here we can conclude $r\in (t^{a})$ and finish the proof.