# Confused about using "the" in mathematical English writing [closed]

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?

• This is a linguistic question, not a mathematical one. The word "the" has multiple meanings and usages: learnenglish.britishcouncil.org/grammar/… Commented May 20 at 12:48
• 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. Commented May 20 at 12:58
• 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. Commented May 20 at 13:29

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.
• 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 $\ldots$") Commented May 20 at 13:54
• 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. Commented May 20 at 21:13
• @user1082245 IMHO, "Ideals of $R$ are ..." could be understood as ""Some ideals of $R$ are ..." Commented May 20 at 21:15