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I am uncertain whether I should use indefinite pronoun "a" or the definite pronoun "the" below. Could someone explain the choice here? I am trying to make a general definition for a principal ideal.

A principal ideal, $I$, of a ring, $R$, is the ideal generated by a single element, $e\in R$, by multiplying it with each element of $R$.

hhh
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  • You should use "a". However, I think instead of "is the ideal generated" you should write "is an ideal generated". – smcc Aug 22 '16 at 15:14
  • @smcc you mean that all pronouns should be indefinite, also an ideal generated by...? – hhh Aug 22 '16 at 15:15
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    @smcc I disagree; the definition is clearer as written, even if both are technically correct. – Ben Grossmann Aug 22 '16 at 15:21
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    Yes, you should say "an ideal generated". @Omnomnomnom There is not just one principal ideal of a ring, and as written the definition seems to imply there is only one principal ideal. – smcc Aug 22 '16 at 15:25

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Definite articles indicate the uniqueness of the object they refer to (within some context). It makes sense to say a ring $R$ because there are many (more than one) rings that could be considered. It makes sense to say "a single element" because in general, rings contain many elements.

It would be strange to say "a multiplicative identity of $R$" because multiplicative identities are unique; there is only one in a given $R$.

Ben Grossmann
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  • What about the ideal generated by...? Is that correct or an ideal? – hhh Aug 22 '16 at 15:21
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    It's correct: a given element generates exactly one ideal. – Ben Grossmann Aug 22 '16 at 15:22
  • But there is not just one principal ideal... – smcc Aug 22 '16 at 15:23
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    @smcc uniqueness is context-dependent. There are many principle ideals. However, given an element $e \in R$, there is exactly one principle ideal $\langle e \rangle$ generated by that element. – Ben Grossmann Aug 22 '16 at 15:25
  • That is irrelevant. Of course I would say "the principal ideal generated by $e\in R$". However, in the definition the element $e$ is introduced after: there are many $e\in R$" to choose to form "a principal ideal". It should be consistent with how the sentence starts: "A principal ideal", "an ideal generated". – smcc Aug 22 '16 at 15:28
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    The later statement about $e$ provides the context for the earlier "the", even though it comes later in the sentence. If you disagree with me about the English language working this way, then we'll just have to agree to disagree. – Ben Grossmann Aug 22 '16 at 15:29
  • @smcc you could try posting a question on the English Language SE, if you want a second opinion. – Ben Grossmann Aug 22 '16 at 15:30
  • @Omnomnomnom Just look up definitions on the internet if you are still so sure I am wrong. Would you really write "A principal ideal is the ideal generated"? – smcc Aug 22 '16 at 15:32
  • @smcc again, my point is not that your version is wrong, it's that it's slightly less clear than it can be, which is at best a matter of opinion (more to the point, the existence of published definitions using your phrasing fails to refute my point). Your point is that I am wrong, which I am not, but feel free to think so. – Ben Grossmann Aug 22 '16 at 15:37
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    @smcc I'll admit that there's something strange about "a principal ideal... the ideal" as opposed to "a principal ideal... an ideal". I can't come up with a grammatical reason, however, that the first is wrong. It seems slightly less strange when it is separated by the comma. – Ben Grossmann Aug 22 '16 at 15:42
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    @smcc how about the following: if $I$ is the ideal generated by a single element $e \in R$, we call $I$ a principal ideal* of $R$. That sounds okay to me, even though I switched from the* to a. – Ben Grossmann Aug 22 '16 at 15:44
  • @Omnomnomnom The problem I have with the definition in the OP is that an/the are not even used consistently when referring to "(principal) ideal". I am not sure why you think changing "the ideal generated" to "an ideal generated" makes things "less clear". Where do you see the amibguity? I think your suggested definition would be better reworded as "If an ideal $I$ is generated...". – smcc Aug 22 '16 at 16:11