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If I understand correctly, this question will be deleted. In my opinion, this would be unfortunate because the answer to it are (I think) of at least as high quality as the answers to the "Atiyah-MacDonald, exercise 2.11" question (mathoverflow.net/questions/136/atiyah-macdonald-exercise-2-11). The best would be of course to append the answers to this question to the answers to the other question. But if this is too complicated, it would be better to reopen this question.
Dear Jacques Carette: David Speyer wrote above: "I don't understand. For every value of $x$ between $0$ and $1$, we have $\sin^{-1}((1-x)^{1/2}) = \cos^{-1}(x^{1/2})$. So take $x$ to be any rational number in this range." What's wrong with this argument?
Dear kwan: Here is how I understand your answer. Lazarus proves the existence of non-equipotent maximal linearly independent subsets of modules over commutative rings. Is this interpretation of your wording correct?
Jacobson's Theorem says that the subset $S$ of $\mathbb Z[X]$ formed by the polynomials $X^n-X$, $n > 1$, has the following property. If, for every $a$ in any given ring $A$, there is an $f$ in $S$ such that $f(a)=0$, then $A$ is commutative. Is $S\cup-S$ maximal for this property?
You wrote "When a group A is isomorphic to a group B and the group G is simple, then we can infer that the group B is also simple." Don't you mean A instead of G?
Dear Harry: Thank you very much for your answer, which confirms my intuition. Would it be indiscreet to ask: What is your personal position on these questions?
