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I'm came across the following proposition which confused me: enter image description here

Is there a typo when both $ I = (g) $ and $ J = (g) $? And if not, why use the same notation to describe an ideal of two different polynomial rings?

Grotto Box
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    $I$ is the ideal generated by $g$ in the ring $F[X]$; $J$ is the ideal generated by $g$ in the ring $R[X]$. – Gerry Myerson Jan 18 '23 at 03:24
  • It's a common abuse of notation to use the same notation in an extension ring when the denotation is clear, like writing an integral fraction as $,r,$ vs. $,r/1,,$ or constant polynomials as $,c,$ vs. $,c x^0,,$ etc. – Bill Dubuque Jan 18 '23 at 03:27
  • Those two ideals are different: the first is "the ideal generated by $g$ in $R[x]$", the second is "the ideal generated by $g$ in $F[x]$" . It is unfortunate that the notation $(g)$ used in the text does not specify the ring, that is why the author was compelled to clarify with $(g)\lhd R[x]$ and $(g)\lhd F[x]$. –  Jan 18 '23 at 03:36
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    Slightly off topic but I'm going to indulge one of my favourite rants. You don't always have to write mathematics using symbols: in some (many?) situations, words are better. Just see how the comment from @GerryMyerson makes the issue crystal clear. – David Jan 18 '23 at 03:36
  • @David One can also argue the opposite: too wordy presentations may obscure key innate mathematical structure, e.g. using words to denote a polynomial would be extremely obfuscational vs. the efficient standard notation $,a_0 + a_1 x + \cdots + a_n x^n.,$ There are often good reasons for abusing notation, e.g. see here for the reason why we use the same tuple notation for gcds and ideals. – Bill Dubuque Jan 18 '23 at 03:52
  • @BillDubuque sure, but as far as I have seen in my experience of reviewing both student and professional work, "too few words" is by far the more common problem. IMHO more words would definitely be an improvement in the present case, where the abuse of notation was clearly confusing the OP. – David Jan 18 '23 at 04:26
  • I trust it's obvious that I am not suggesting the best way to solve a quadratic is to take the negative of the linear coefficient, plus or minus the square root of the square of the linear coefficient minus the product of the quadratic and constant coefficients, divided by twice the quadratic coefficient. – David Jan 18 '23 at 04:29
  • Ok thanks so much for the insight everyone! I'm new to this subject and the author has had occasional typos so I just wanted to make sure. I appreciate it! – Grotto Box Jan 18 '23 at 18:53

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