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Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$.

I can't figure out where to start other that I need to prove some biconditional statement. Any help?

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Think about what happens to ideals when they contain a unit. –  sxd Jan 21 '12 at 23:21

4 Answers 4

up vote 13 down vote accepted

From some of the comments above you seem a little confused. Since you said you are not familiar with proofs I will try to write this out in a way that you can understand.

You are trying to prove the equivalence of the following statements:

$P:$ A commutative ring $R$ with $1$ is a field.

$Q:$ The only ideals of $R$ are $(0)$ and $(1)$.

Let us look at statement $Q$ closely. Well it is saying that the only ideals of $R$ are the zero ideal (which has only one element, zero) and $(1)$. What is $(1)$? Well by definition of an ideal if you multiply anything in the ideal $(1)$ by anything in $R$, you should get something back in $(1)$ again. But then $1$ is the multiplicative identity of $R$ so multiplying everything in $R$ by $1$ just gives everything in $R$ back again. This means that $(1)$ must be the whole ring $R$.

Now suppose we want to prove $P \implies Q$. Let $I$ be an ideal of a ring $R$. Here "ring" means "commutative ring with a unit". Now here are some things you should know:

(1) $I$ is non-empty

(2) $I$ must at least contain the element $0$ (Why?)

(3) If $I$ has more than one element this means that at least one non-zero element $a$ of the ring must be in $I$. (Why?)

Therefore if $I$ contains only $0$, $I = (0)$. If $I$ does not only contain zero, then by (3) above it contains at least one non-zero element $a$ of $R$. Now recall that we are trying to prove $P \implies Q$. We already know $P$. Therefore this means by definition of a field that $a^{-1}$ exists in $R$.

But then by definition of an ideal $I$, $a^{-1} a = 1$ must be in $I$. Therefore $1 \in I$ so that $I$ must be the whole ring $R$. Hence $I = (1)$. This establishes $P \implies Q$.

Now for the converse:

To show $Q \implies P$ it suffices to show that non-zero every element $a \in R$ contains a multiplicative inverse. So let $a$ be a non-zero element of $R$. The trick now is to consider the principal ideal generated by $a$ (which we denote by $(a)$ ).

Now by assumption of $Q$, since the only ideals of $R$ are $(0)$ and $(1)$, this means that $(a)$ being an ideal of $R$ must be either $(0)$ or $(1)$. Now $(a)$ cannot be $(0)$ for $a \neq 0$. So $(a) = (1)$. But then this means that $1$ is a multiple of $a$, viz. there exists a non-zero $c$ such that

$$ac = 1.$$

However this is precisely saying that $a$ has a multiplicative inverse. Since $a$ was an arbitrary non-zero element of $R$, we are done. Q.E.D.

Does this help you? I can discuss more if you need help.

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this is very detailed explaination and very useful. –  zinking Oct 7 '12 at 5:48

Let $a\in R$, $a\neq 0$. Then the smallest ideal of $R$ that contains $a$ is $Ra=\{ra\mid r\in R\}$.

(Verify that the set $\{ra\mid r\in R\}$ is an ideal of $R$, and that it contains $a$; it's not hard. Then think about why any ideal that contains $I$ must contain this set.).

If $R$ is a field, then what is $Ra$ for any $a\neq 0$? If $I$ is any ideal of $R$, and $a\in I$ with $a\neq 0$, what does that tell you?

Conversely, if the only ideals of $R$ are $(0)$ and $R$, what is $Ra$? What does that tell you about $a$?

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Since the asker had no idea where to start, I suspect they may not know that $Ra$ is an ideal. Drew sam, can you prove that? –  user23211 Jan 21 '12 at 23:05
    
Ummm, for the first part, I am not sure what you mean. You just multiply the field by that value? Is it just the identity? –  Jackson Hart Jan 21 '12 at 23:07
    
And for the second part, is a just e? –  Jackson Hart Jan 21 '12 at 23:08
    
@Drew, I don't know what you mean by "e". Arturo has written in his answer what he understands by $Ra$ and by $a$. Just read it carefully. –  user23211 Jan 21 '12 at 23:12
3  
@drew sam: If that's the case, then let me suggest that this website and quick interaction through comments is not an appropriate forum for you; you should try to find one-on-one, personal help at your institution instead, or try to think through things before posting quick follow-up questions as you have done here. –  Arturo Magidin Jan 21 '12 at 23:37

Hint $\ $ field $\rm R\ \iff\ 0\:|\:r\ \ or\ \ r\:|\:1 \iff\ (0)\supset (r)\ \ or\ \ (r)\supset (1)\iff\ (0)\supset I\ \ or\ \ I\supset (1)$

Alternatively it is the special case $\rm\: I = 0\:$ of: $\, $ field $\rm\: R/I \iff I\:$ maximal.

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Do you mean that a is in r? –  Jackson Hart Jan 21 '12 at 23:21
    
@drew I see no "a" above. r denotes any element of R. –  Bill Dubuque Jan 21 '12 at 23:24

It is given that the $0$-ring $\{0\}$ and the whole ring $R$ are ideals of any ring $R$. (Here I assume $R = \mathbb{Q}$, the rationals)

Suppose by contradiction that there is another ideal $N$ of $Q$ that is not $\{0\}$ or all of $Q$.

$N \neq \{0\}$ implies there is a nonzero rational number $a$ contained in $N$. Since $\frac{1}{a}$ is contained in $\mathbb{Q}$, the ideal $N$ must contain $(\frac{1}{a})\cdot a = 1$.

$N \neq Q$ implies that there is a rational number $b$ which is not contained in $N$. But $N$ contains $1$, so $N$ must also contain $b\cdot 1 = b$ by the definition of an ideal, contradicting the assumption that $N$ is not all of $\mathbb{Q}$.

Therefore there are no proper, non-trivial ideals of $\mathbb{Q}$.

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For some basic information about writing math at this site see e.g. here, here, here and here. –  Julian Kuelshammer Nov 2 '12 at 19:57
    
Also this answers only one direction of the question (only for $\mathbb{Q}$, but that generalises easily). –  Julian Kuelshammer Nov 2 '12 at 20:00

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