I'm new to graded rings and am having a bit of trouble figuring this problem out. I am trying to figure out the homogeneous prime ideals of $R=k[x,y]/(x^2y)$, where $x$ and $y$ have degree $1$ which don't contain the irrelevant ideal. The irrelevant ideal is given by $R^{+}=\bigoplus R_n$ for $n\geq 1$, whre $n$ is the grading of $R$. I have tried to proceed as follows:
I think (by considering the prime ideals of k[x,y] that contain $(x^2y)$) the prime ideals of $R$ are $(0), (x), (y), (x,y-a), (x-a,y)$, for $a\in k$.
I thus have two questions/requests for verification:
(1) The homogeneous ideals are $(x), (y), (0)$, right? As $x-a$ and $y-a$ are not homogeneous.
(2) Do $(x)$ and $(y)$ contain $R^+$? $R^+$ is a bit hard for me to calculate here...
Thank you for any help, I feel as if I'm overlooking something in this math problem.
Edit: Also please let me know if the prime ideals are somehow wrong when considering graded rings.
Edit2: As pointed out in AlexL's answer, $(x,y)$ is also a homogeneous prime ideal, but it contains $R^{+}$
Here $R^+$ is just the ideal generated by $x$ and $y$ : $R^+=(x,y)$. Writing the grading $k[x,y]= \bigoplus_n A_n$ ($A_n$ the vector space of degree $n$ homogeneous polynomials), then $R=\bigoplus R_n$ for $R_n=A_n/(x^2yA_{n-3})$ (take $A_{-3}=A_{-2}=A_{-1}=0$)
$(x,y-a)$ and $(x-a,y)$ are not homogeneous for $a \neq 0$ (otherwise $a$ should be in the ideal and so the ideal would be whole $R$), but $(x,y)$ is homogeneous and maximal