In an undergraduate number theory class, I was taught the Chinese Remainder Theorem as follows: where $n_1,\cdots,n_k$ are pairwise coprime integers and $\{a_i\}\subseteq \mathbb Z$, the set of equivalences $x\equiv a_i\ \text{mod}(n_i)$ for $i=1,\cdots,k$ has a solution that is unique modulo $\prod_{i=1}^k n_i$.
Now that I am studying Dummitt and Foote's text in graduate school, I get their version of the CRT: Let $A_1,\cdots,A_k$ be ideals of a ring $R$. The map $$R\to R/A_1\times \cdots \times R/A_k$$ defined by $r\mapsto(r+A_1,\cdots,r+A_k)$ is a ring homomorphism with kernel $\cap_{i=1}^k A_i$. If these ideals are pairwise comaximal, then this map is surjective and $\cap_{i=1}^kA_i=A_1A_2\cdots A_k$, so $$ R/(A_1A_2\cdots A_k)= R/(\cap_{i=1}^k A_i)\cong R/A_1\times \cdots \times R/A_k $$
I cannot figure out how to reconcile these two versions. More specifically, my question is: Given the latter formulation, how can I show the former formulation to be true?