What does "with respect to the exclusion of a set" mean?

Question

I'm not good at English, but I'm studying math in English. Can anyone explain to me what is this sentence means?

Let $Q$ be an ideal maximal with respect to the exclusion of $T$.

Thank you.

Answer 1

It is an ideal that is the largest without containing any elements of T. That is to say all other ideals that do not contain elements of T are subsets of Q.

Answer 2

As an alternative to @Alephnull's intepretation, maybe it's a mistake and the sentence should have been

let $Q$ be a maximal ideal in $T$ with respect to inclusion.

If the sentence has been written by a non-native English speaker, this intepretation seems reasonable.

Answer 3

If you are studying multiplicative sets, it is probably in connection with this lemma:

If $R$ is a commutative ring, $T$ is a multiplicative subset$^\ast$ and $I$ is an ideal maximal with respect to exclusion of $T$, then $I$ is a prime ideal.

It means that $I$ has the property that if $J$ is another ideal properly containing $I$, then $J\cap T\neq \emptyset$.

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$^\ast$ the definition ought to include the clause that $0\notin T\neq\emptyset$, of course