About module decomposition in Atiyah-Macdonald

Question

I am reading p. 20 of Introduction to Commutative Algebra by Atiyah and Macdonald. Given a module decomposition

$$ A=\mathfrak{a}_1\oplus\cdots\oplus\mathfrak{a}_n $$ of ring $A$ as direct sum of ideals $\mathfrak{a}_i$ What does module decomposition mean in this line?

1. Internal direct sum
2. External direct sum
3. $\mathfrak{a}_i \cap \mathfrak{a}_j=0$ for $i\neq j$

Answer 1

This means internal direct sum. By definition, $\mathfrak{a}_i\cap \mathfrak{a}_j=0$ for $i\ne j$ and $\mathfrak{a}_1+\cdots+\mathfrak{a}_n=A$.