Is logical "or" exclusive or inclusive in prime ideal definition

Ok, so an ideal is prime if for every $ ab \in I $, $ a \in I $ or $ b \in I $.

Now the two questions:

1: Is that an inclusive or exclusive or? Can both $a$ and $b$ be in I?

2: Can $ab$ be equal to either $a$ or $b$?

Solution 1:

1. You can take as a point of universal convention of mathematics that "A or B" always means "A inclusive or B". Some times "A and B" is impossible (for instance, if you throw a die and you're interested in the probability of a 5 or a 6), but even in those cases, the "or" is inclusive (we know it is impossible for both a 5 and a 6, but that's because we know how a die works, not because we have changed the meaning of "or").

2. Yes. If $a=1$, then $ab=b$, and vice versa. For instance, in the ring of integers, we have the prime ideal $(5)$. One manifestation of its primeness is that since $1\cdot 10\in (5)$, we do indeed have $1\in (5)$ or $10\in (5)$.