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I have some questions regarding graded ideals.

Firstly, what is the definition of a graded ideal? Is it the same definition as graded ring, where the ideal is viewed as a subring?

Secondly, for a polynomial ring over a field $F$, $F[t]$, why are its graded ideals all of the form $(t^n)$?

What I know is $F[t]$ is a Euclidean domain, and hence a PID.

Thanks for any help.

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    This may help you: https://ysharifi.wordpress.com/2010/12/18/graded-algebras-basic-definitions-and-examples/ and https://ysharifi.wordpress.com/2011/04/22/graded-ideals/ – 2017-02-10

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Let $R=\oplus_{g\in G} R_g$ be a $G$-graded ring ($G$ a commutative monoid). An ideal $I$ of $R$ is said to be a graded ideal, if $I=\oplus_{g\in G} (I \cap R_g)$. It turns out that $I$ admits a generator consisting of homogeneous elements of $R$ (see e.g. Proposition 2.1 of this nice note).

Concerning the question about the graded ideals of $F[t]$, it is enough to note that $F[t]$ is a PID, whenever $F$ is a field. So if $I$ is a graded ideal of $F[t]$, then it follows from definition that $I$ has a homogeneus generator(s). Note that the only homogeneous elements of $F[t]$ (viewd as $\mathbb{N}$-graded ring) are $t^a, a \in \mathbb{N}$.