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I am not an English native speaker. In many times, I have seen different authors used "the" or not used "the" in the same situation. I am so confused about this. Here are some examples. The first example,

  • ... operations on the ideals of $R$ ...
  • ... operations on ideals of $R$ ...

The second example,

  • ... the definition of the xxx operation ...
  • ... the definition of xxx operation ...

The third example,

  • ... all the ideals of $R$ ...
  • ... all ideals of $R$ ...

Could anyone tell me which one in the above is right or give some other explanations?

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    $\begingroup$ This is a linguistic question, not a mathematical one. The word "the" has multiple meanings and usages: learnenglish.britishcouncil.org/grammar/… $\endgroup$
    – freakish
    Commented May 20 at 12:48
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    $\begingroup$ Both are correct, though I couldn't explain why or whether there's any slight semantic difference. For that, you might want to ask on the Linguistics or English Language stack exchange. $\endgroup$
    – Sambo
    Commented May 20 at 12:58
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    $\begingroup$ Mathematics uses lots of words and constructions idiosyncratically, as part of math jargon, (that's jargon in the sense of "specific to a field", not with the sense of "slang"), so I think this is a valid question to ask mathematicians. But sadly I think you'll just get opinions, which will put this outside of the permitted questions range. $\endgroup$
    – JonathanZ
    Commented May 20 at 13:29

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As far as the math is concerned, I believe the examples you mentioned are the same. But you must be careful about the distinctions between "the" and "a/an" in contexts of definitions, and uniqueness. For example, "the ideal of $R$" carries the implication that $R$ is a ring with exactly one ideal, whereas while referring to "an ideal of $R$", no such property of $R$ is implied.

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  • $\begingroup$ (+1) for the "the" and "a/an" distinction you make. I've explained it this way to prospective ESL writers of mathematics questions for certain high-stakes graduate program admissions tests: Use “the” when making reference to a unique object, such as a specific object previously mentioned. Use “a/an” when making reference to one of possibly several objects. Interestingly, this distinction is almost always ignored, even by native English speakers, (continued) $\endgroup$ Commented May 20 at 13:54
  • $\begingroup$ when referring to a proof, such as: "The proof of this assertion is given below, but first we mention $\ldots$" (instead of "Our proof $\ldots$" or "A proof $\ldots$") $\endgroup$ Commented May 20 at 13:54
  • $\begingroup$ @DaveL.Renfro Why not active rather than passive voice? "We prove this assertion below, but first ...". It's less wordy and bypasses "The" vs "A" vs "Our" proof. $\endgroup$ Commented May 20 at 21:10
  • $\begingroup$ Could you tell me which one in the following I should use at the beginning of a sentence? "The ideals of $R$ are one the the following forms..." or "Ideals of $R$ are one of the following forms...". Should I always add "the" like "the ideals of $R$" to ensure it sounds correct? It is hard for me to tell the differences. $\endgroup$ Commented May 20 at 21:13
  • $\begingroup$ @user1082245 IMHO, "Ideals of $R$ are ..." could be understood as ""Some ideals of $R$ are ..." $\endgroup$ Commented May 20 at 21:15

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