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  1. Let $S$ be a multiplicatively closed subset of a ring $R$, and let $I$ be an ideal of $R$ which is maximal among ideals disjoint from $S$. Show that $I$ is prime.

  2. If $R$ is an integral domain, explain briefly how one may construct a field $F$ together with a ring homomorphism $R\to F$.

  3. Deduce that if $R$ is an arbitrary ring, $I$ an ideal of $R$, and $S$ a multiplicatively closed subset disjoint from $I$, then there exists a ring homomorphism $f\colon R\to F$, where $F$ is a field, such that $f(x)=0$ for all $x\in I$ and $f(y)\neq 0$ for all $y\in S$.

[You may assume that if $T$ is a multiplicatively closed subset of a ring, and $0\notin T$, then there exists an ideal which is maximal among ideals disjoint from $T$.]

Here is a question I need to answer. For the first part I can show if $x, y$ are not in $I$ then nor does $xy$. The second part is just the field of fractions.

For the third part, I think I need to find an ideal $J$ containing $I$ such $J$ is prime, so that $R/J$ is an integral domain, and use the second part. To find $J$ prime, I think as suggested by the hint, I should go for a maximal ideal disjoint from $S$ containing $I$, but how can I do that?

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    Wait, please slow down. You just asked a question that turned out to be trivially false, and some people there (including me!) have been putting some effort into trying to figure out what you meant to ask. Could you please address this before moving on to a new question?2012-05-20
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    In fact, the first question you ask is answered in a link I gave in a comment to your last question. This link was provided based on educated guesswork by @Benjamin Lim and myself on what you might have meant. So again, please slow down. Also, you have reproduced verbatim what looks to be a portion of a problem set. *Is* this homework of some sort? What is the source you are quoting from?2012-05-20
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    @PeteL.Clark This is from a set of practice questions given to us for exam preparation.2012-05-20
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    @Zhang: thanks for the clarification. I don't really feel comfortable helping you solve practice problems for an exam. I don't find it ethically problematic *per se*, but I strongly recommend that you discuss them with your instructor and/or your classmates first. If you cannot do the practice problems, that's useful information for the instructor to have. Besides, getting other people to solve your practice problems may not be such good practice.2012-05-20
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    @BenjaminLim Yes. If $x, y$ are outside $I$ then by maximality, $(x, I), (y, I)$ both intersect $S$. So for some $r, s\in R$, $i, j\in I$, $xr+i, ys+j\in S$. So $(xr+i)( ys+j)\in S$, and on expanding one sees $(xy, I)$ intersects $S$, so $xy$ is outside $I$. Hence $I$ is prime.2012-05-20
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    @Zhang Ok now for your second problem can you do it?2012-05-20
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    @PeteL.Clark Thanks for the considerations. I have discussed with other students but with no results, they couldn't reproduce the supervisor's solution. I missed my supervisor appointment. That's why I am asking here.2012-05-20
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    @PeteL.Clark I have given the OP some hints for (3) which he should be able to complete from here.2012-05-20

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