Everywhere I read it's "a is element of I or b is element of I", but I couldn't find an explanation if it's a mathematical or (so that it can be the case that a is element of I and b is element of I), or if it's a language or (so that it's impossible that both belong to I)
2026-04-06 20:36:39.1775507799
Prime Ideals: $ab\:\in \:I$ such that $a\:\in \:I\:\:\vee \:\:b\in I$ vs. either $a\:\in \:I\:$ or $b\:\in \:I\:$ but not both?
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It's the former - inclusive not exclusive or. In math, we always use the inclusive or unless explicitly stated otherwise.
As a sanity check, your other definition doesn't really even make sense. Take an ideal $I \subseteq R$ that is "prime" per your second definition, and take $a, b \in I$. Then by definition of an ideal, $ab \in I$. But by this exclusive or definition, precisely one of $a, b$ can be in $I$. In other words, no ideals are "prime" per your second definition.