Wrong proposition in "Atiyah and Macdonald"s book?!

Question

In page 6 of "Introduction to commutative algebra" write that:

$a \cap b = ab$ provided $a + b = (1)$

But I think it's not true, by considering $a = b = (2) \in \mathbb Z_6$

Answer 1

The proposition can be reformulated as

if $\mathfrak{a}+\mathfrak{b}=(1)$, then $\mathfrak{a}\cap\mathfrak{b}=\mathfrak{a}\mathfrak{b}$

The word “provided” is used in the sense of “when it is given that”.

In your case $\mathfrak{a}+\mathfrak{b}=(2)+(2)=(2)\ne(1)$.