# Homogeneous and maximal ideal in a $\mathbb Z$-graded ring

Is Exercise 2.8 from Marley's notes on "GRADED RINGS AND MODULES" true?

Exercise 2.8: Let $R$ be a graded ring and $M$ a homogeneous maximal ideal of $R$. Prove that $M =…⊕R_{-1}⨁m_0⨁R_1⨁…$, where $m_0$ is a maximal ideal of $R_0$.

I mean is $…⊕R_{-1} ⨁m_0⨁R_1 ⨁…$ necessarily an ideal?

• guess the confusion comes from the word "homogeneous maximal ideal". If it means that it is homogeneous and maximal then what @user121097 described can not happen. But if it meant maximal among homogeneous ideal, then it is false in general. Mar 7, 2014 at 21:03
• @Youngsu I don't understand your comment: the question is asking whether $\cdots \oplus R_{-1}\oplus m_0\oplus R_1\oplus\cdots$ is an ideal in a $\mathbb Z$-graded ring $R$. Is it? Mar 8, 2014 at 10:36
• @user121097: No, in general. Yes, if homogeneous maximal means $M$ is homogeneous ideal and $M$ is maximal. $\;\;$First of all, a maximal homogenous ideal (an homogeneous ideal which is maximal with respect to inclusion) is not unique. If this happens then you don't even get $R_0$ local. Secondly, say $M$ is unique maximal homogeneous ideal. Then $R_0$ is local, but your example still works. Take $R = k[t,t^{-1}]$. Then the maximal homogeneous ideal is $(0)$ and of course $t * t^{-1} = 1 \notin M$. Observe that $R / M = R/(0) = R$ which is not a field. So $M$ is not maximal. Mar 8, 2014 at 19:24
• The problem here is that $R$ has a unit of positive degree. Lastly, if $M$ is homogeneous and maximal, then $R$ does not have a unit of positive degree. Therefore, the example you presented does not occur in this case. Furthermore, this says that $R_i$ is in $M$ if $i \neq 0$ since they do not contain a unit and they are homogeneous. Mar 8, 2014 at 19:25

1) One is asking if $\cdots\oplus R_{-1}\oplus m_0\oplus R_1\oplus\cdots$ is an ideal in a $\mathbb Z$-graded ring $R$, where $m_0$ is a maximal ideal of $R_0$, and the answer is negative as shows the following example: $R=k[t,t^{-1}]$, and (necessarily) $m_0=(0)$.
2) The other is an exercise in Marley's notes asking to prove that any homogeneous and maximal ideal $M$ in a $\mathbb Z$-graded ring $R$ has the form $M=\cdots\oplus R_{-1}\oplus m_0\oplus R_1\oplus\cdots$, where $m_0$ is a maximal ideal of $R_0$. In this case $R/M$ is a graded ring and also a field. This shows that $(R/M)_n=0$ for all $n\ne 0$ (why?), that is, $R_n=M\cap R_n$ for all $n\ne 0$. If $m_0=M\cap R_0$ it follows that $M=\cdots\oplus R_{-1}\oplus m_0\oplus R_1\oplus\cdots$.
• I pretty much know nothing about graded rings. So, could you clarify why the graded field is necessarily concentrated in degree 0 ? this is obvious for "\mathbb{N}"-graded situation. For instance $k[t,t^{-1}]$ is graded field. Right ? Oct 27, 2018 at 14:53