Maximal and Prime Ideals in a finite Ring of Functions

Asked Jun 23 '17 at 20:08

Active Jun 23 '17 at 20:35

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I was trying to solve a question asked in some PhD entrance exam but I got stuck in some options. The question is:

Let $R$ be the set of all functions from the set $\{1,2,...,10\}$ to $\mathbb{Z}_2$. Then $R$ is a commutative ring with pointwise addition and pointwise multiplication of functions. Then which of the following statements are true?

$R$ has a unique maximal ideal.

Every prime ideal of $R$ is maximal.

Number of Proper ideals of $R$ is $511$.

Every element of $R$ is idempotent.

Now it is obvious that statement $4$ is correct.

And I found that option $1$ is false because for each $a \in X$, the set of all functions which maps $a$ to $0$ forms a maximal ideal (consider the kernel of evaluation at $a$, i.e. the kernel of the map $T$ from $R$ to $\mathbb{Z}_2$ sending $f$ to $f(a)$. From here we get that $R/\ker T$ is isomorphic to $\mathbb{Z}_2$ and hence the ideal will be maximal). So, we get at least $10$ maximal ideals.

But I could not get any hint in proving or disproving options $2$ and $3$.

Any help would be appreciated.

edited Jun 12 '20 at 10:38

Community

asked Jun 23 '17 at 20:08

Sumit Mittal

1

It's a finite ring; surely you know that every finite integral domain is a field? – Angina Seng Jun 23 '17 at 20:10
1

its not an integral domain. – Sumit Mittal Jun 23 '17 at 20:12
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@carmichael561 It's a heavy hint for part 2. – Angina Seng Jun 23 '17 at 20:14
Ah, I see what you're getting at. – carmichael561 Jun 23 '17 at 20:15
Hint (for #2): If $I$ is a prime ideal, then $R/I$ is integral domain. But since $R$ is finite, $R/I$ is a finite integral domain. What do you know about finite integral domains? – TASPlasma Jun 23 '17 at 20:52
$511$ is a trap ? Or did I miss something. – reuns Jun 23 '17 at 20:57
Please search for your keywords before posting next time, and you are likely to find the exact duplicate that already existed. – rschwieb Jun 23 '17 at 21:39

Maximal and Prime Ideals in a finite Ring of Functions

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