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I have the following question in a competitive exam , but I failed to answer it.The question is:

Let $\mathcal R = \{f:\{1,2, \dots , 10\} \rightarrow \mathbb Z_2\} $ be the set of all $\mathbb Z_2$-valued functions on the set $\{1,2,\dots , 10\}$. Then $\mathcal R$ is a commutative ring with pointwise addition and multiplication of functions.Which of the following statements are correct?

  1. $\mathcal R$ has a unique maximal ideal.
  2. Every prime ideal of $\mathcal R$ is also maximal.
  3. Number of proper ideals of $\mathcal R$ is 511.
  4. Every element of $\mathcal R $ is idempotent.

The only option I was able to answer was option 4 .

Can anyone help me understanding the ring in the question and the options.

Any insight will be happily appreciated. Thank you.

hiren_garai
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    This ring is isomorphic to the direct product of 10 copies of $\mathbb{Z}_2$. This could make some of this simpler. – Randall Sep 03 '17 at 15:45
  • What commands you used to edit the question,?Actually I am pretty new to Latex , will you help me with some tips .. – hiren_garai Sep 03 '17 at 15:47
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    @HirenGarai You can check the edit by clicking on "edit". Also: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference. – anderstood Sep 03 '17 at 15:55
  • That's pretty useful link.Thanks @anderstood. – hiren_garai Sep 03 '17 at 15:57
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    You might try your hand at editing by improving the title. While it makes sense to reference the source of the problem in the body of the Question, the title will be better used to alert Readers to the nature of the problem. – hardmath Sep 03 '17 at 16:11
  • I hope this one is good enough , right ?@hardmath – hiren_garai Sep 03 '17 at 16:34
  • @HirenGarai Please search for your question before asking it. It is currently the fourth exact duplicate of this same question. That may change if some of the duplicates get deleted. – rschwieb Sep 04 '17 at 00:07
  • Ok. I will keep in mind that in future.@rschwieb – hiren_garai Sep 04 '17 at 00:17

1 Answers1

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Hint for 1: The ideal consisting of functions $f$ such that $f(3)=0$ is maximal; is $3$ special?

Hint for 2: What's a finite domain?

Hint for 3: I bet on 1023, because $2^{10}=1024$.

Statement 4 is indeed true, because every element in the two element ring is idempotent.

egreg
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