For a ring $A$ and an ideal $\mathfrak{a}$ of $A$, Atiyah-Macdonald define $$A^*=\bigoplus_{n=0}^\infty \mathfrak{a}^n$$ and claim that it is a graded ring on p. 107 of their commutative algebra book. My confusion is that I cannot see how $A^*$ is a ring. Does the direct sum represent the internal direct sum or the external direct sum?
Since $\mathfrak{a}^0\supseteq\mathfrak{a}^1\supseteq\cdots$, I think the internal direct sum is not possible. If it is the external direct sum, then what would be the multiplicative identity of $A^*$? Maybe I am missing something completely obvious.
It's the external direct sum, yes (as you remark the internal direct sum wouldn't make sense). The product is such that $\mathfrak{a}^i \cdot \mathfrak{a}^j \subset \mathfrak{a}^{i+j}$. The unit is the unit of $A = \mathfrak{a}^0$.