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So I'm trying to prove that if $M$ is a maximal ideal in $R$ such that for all $x\in M$, $x+1$ is a unit, then $R$ is a local ring with maximal ideal $M$, that is to say $R$ has a unique maximal ideal.

I've been at this one for a day now and I just can't figure it out. I have that $R$ being a local ring is equivalent to there being a proper ideal $I$ of $R$ which contains all non-units of $R$, and also equivalent to the set of non-units of $R$ being an ideal.

The set of $x+1$ for $x\in M$ is itself multiplicative, but I'm not sure where to go with that since inverting that set just gives back $M$ (since they're all units). I haven't been successful at proving anything about elements of $R$ which are not either in $M$ nor of the form $x+1$ for $x\in M$.

I also tried just assuming there was some other maximal ideal $N$ and then trying to draw out a contradiction by looking at the ideal $M+N$, clearly if $M+N$ doesn't contain $1$ then I've got my contradiction, but I don't seem to have enough information to pursue that path.

Can anyone give me some guidance? Thanks.

Set
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    Take the answer-edit out of your question and put it in a proper answer where it belongs, so other people can see your question is answered. – Ben Millwood Aug 10 '12 at 22:03

5 Answers5

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Actually upon writing this up I believe I've solved it:

Assume there existed another maximal ideal $N$. Then if $1\in M+N$ then there exists $1 = m+n\in M+N$ and thus $n = -m+1$ and since $m\in M$ implies $-m\in M$, this means that $n = -m+1$ is a unit and thus $N=R$. Therefore $M+N$ must not contain $1$ and thus we have found a proper ideal of $R$ which contains $M$, contradicting the fact that $M$ is maximal.

Set
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    Sometimes, that is all you have to do. Lay all the details out so you can have a bird's eye view of the problem. – Arkady Aug 10 '12 at 22:08
  • @FortuonPaendrag: You're absolutely right, this definitely isn't the first time this has happened. – Set Aug 10 '12 at 22:10
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You can find the answer in Atiyah-Macdonald. I'll repeat it here:

Two steps:

  1. If every $y \in R-M$ is a unit, then $R$ is a local ring. In fact every ideal $\neq (1)$ consists of non-units, hence is contained in $M$.

  2. In this case, let $y\in R-M$, we show that $y$ is a unit. Since $M$ is maximal, $(y) + M = (1)$, so there exists $r\in R$ and $x\in M$, such that $ry-x=1$, so $ry=1+x$ is a unit, then $y$ itself is a unit.

fmoura2005
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  • How exactly do you conclude the last step? Say, $ry$ is a unit, then we know that there exists some $z$ such that $ryz=1$, so we know that $z=(ry)^{-1}$. How do we know that there exist $r^{-1}$ and $y^{-1}$ such that $z=y^{-1}r^{-1}$ (which would imply) that $yy^{-1}=1$, i.e. that $y$ is a unit? – exchange Jul 06 '18 at 07:04
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Given $x\in M$, the hypothesis implies that $1+xr $ is invertible for every $r\in R$.
Hence $x\in Jac(R)$, the Jacobson radical of $R$ (Atiyah-Macdonald, Proposition 1.9), so $M\subset Jac(R)$
But the Jacobson radical is by definition the intersection $Jac(R)=\bigcap M_i$ of all the maximal ideals $M_i$ of $R$.
Thus we have $M\subset\bigcap M_i $, which immediately implies that $M$ is the sole maximal ideal of $R$ .

Keenan Kidwell
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Georges Elencwajg
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Hint $\ $ Put $\rm\,J = \it M\ $ in $(2\Rightarrow 1)$ below, yielding $\rm\:1+{\it M}\, \subset U\:$ $\Rightarrow$ $\, M\,$ lies in every maximal ideal, therefore $\,M\,$ is the only maximal ideal.

Theorem $\ $ TFAE in ring $\rm\:R\:$ with units $\rm\:U,\:$ ideal $\rm\:J,\:$ and Jacobson radical $\rm\:Jac(R)\:.$

$\rm(1)\quad J \subseteq Jac(R),\quad $ i.e. $\rm\:J\:$ lies in every max ideal $\rm\:M\:$ of $\rm\:R\:.$

$\rm(2)\quad 1+J \subseteq U,\quad\ \ $ i.e. $\rm\: 1 + j\:$ is a unit for every $\rm\: j \in J\:.$

$\rm(3)\quad I\neq 1\ \Rightarrow\ I+J \neq 1,\qquad\ $ i.e. proper ideals survive in $\rm\:R/J\:.$

$\rm(4)\quad M\:$ max $\rm\:\Rightarrow M+J \ne 1,\quad $ i.e. max ideals survive in $\rm\:R/J\:.$

Proof $\: $ (sketch) $\ $ With $\rm\:i \in I,\ j \in J,\:$ and max ideal $\rm\:M,$

$\rm(1\Rightarrow 2)\quad j \in all\ M\ \Rightarrow\ 1+j \in no\ M\ \Rightarrow\ 1+j\:$ unit.

$\rm(2\Rightarrow 3)\quad i+j = 1\ \Rightarrow\ 1-j = i\:$ unit $\rm\:\Rightarrow I = 1\:.$

$\rm(3\Rightarrow 4)\ \ \ $ Let $\rm\:I = M\:$ max.

$\rm(4\Rightarrow 1)\quad M+J \ne 1 \Rightarrow\ J \subseteq M\:$ by $\rm\:M\:$ max.

Bill Dubuque
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If we're talking in commutative algebra terms where we do have multiplicative identity and commutativity guaranteed in every ring, then I suppose I could add a bit to the fmoura2005's answer. (Following Atiyah-Macdonald Introduction To Commutative Algebra book.) So we know that $ry$ is a unit and therefore invertible, which means there is some $w$ for example such that $w(ry)=1$ and $(ry)w=1$. Now we have $(wr)y=1$ (associativity) which means $y$ is a unit and of course $y(wr)=1$ (by commutativity). So $y$ is a unit and it is from $R-M$ and therefore, using the first part of fmoura2005's answer, we get that $R$ is a local ring.

Giovanna Nova
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