Definition of irreducible - inclusive or exclusive or?

Question

I just met the definition of an irreducible element:

Let $R$ be a commutative ring with identity and let $a \in R$.

We say that a is irreducible if the following all hold:

• $a \neq 0$
• $a \notin R^*$
• $a=bc$ for $b,c \in R \Rightarrow$ $b$ or $c$ is a unit

I was wondering whether the third condition is an inclusive or i.e. could it be that both $b$ and $c$ are units?

Thanks in advance!

Answer 1

If any author ever writes something where they mean exclusive or, but don't specify that it is exclusive, that would be considered at best a typo, and at worst misleading. "Or", by itself, should always mean inclusive or.

In this particular case, however, it doesn't matter, as both of them being units breaks the second point.