If a sum of ideals is the unit ideal, then 1 can be written as a finite sum of elements?

Question

When I read GTM211, I was confused about this question :if a sum of ideals is the unit ideal, then 1 can be written as a finite sum of elements

Answer 1

Each element of a sum of ideals is a (finite) sum of elements, each element from a distinct ideal in the sum.

Thus, if a sum of ideals is $(1)$, then since $1\in (1)$, it follows that $1$ can be so expressed.

Answer 2

I think you are considering unitary ideals. If a unitary ideal is a sum of ideals, then your unitary ideal has the unity (1) and since it is the sum of ideals, every element of it can be expressed as sum of elements of each ideal that is part of the sum, so you can do what you say.