Suppose that $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ are maximal ideals of a ring $R$.
Then $\mathfrak{p}_i+\mathfrak{p}_j=R$ with $i\neq j$ and $\mathfrak{p}_i^a+\mathfrak{p}_j^b=R$ with $a,b$ positive integers.
By the Chinese remainder theorem $$R/\mathfrak{p}_1^{a_n}\cdots \mathfrak{p}_n^{a_n}\approx R/\mathfrak{p}_1^{a_1}\oplus \cdots \oplus R/\mathfrak{p}_n^{a_n}$$
with $a_i$ positive integers.
My question is:
Under this isomorphism the image of the ideal $\mathfrak{p}_1^{b_1}\cdots\mathfrak{p}_n^{b_n}/\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$ is $\mathfrak{p}_1^{b_1}/\mathfrak{p}_1^{a_1}\oplus\cdots\oplus \mathfrak{p}_n^{b_n}/\mathfrak{p}_n^{a_n}$ for positive integers $b_i\leq a_i$?
It is clear that the image of the ideal $\mathfrak{p}_1^{b_1}\cdots\mathfrak{p}_n^{b_n}/\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$ is a subset of $\mathfrak{p}_1^{b_1}/\mathfrak{p}_1^{a_1}\oplus\cdots\oplus \mathfrak{p}_n^{b_n}/\mathfrak{p}_n^{a_n}$.
But the other inclusion not is clear for my.
Thank you all.
Note: The isomorphism in the Chinese Remainder Theorem is:
$$R/\mathfrak{p}_1^{a_n}\cdots \mathfrak{p}_n^{a_n}\approx R/\mathfrak{p}_1^{a_1}\oplus \cdots \oplus R/\mathfrak{p}_n^{a_n}, \ x+\mathfrak{p}_1^{a_n}\cdots \mathfrak{p}_n^{a_n}\mapsto (x+\mathfrak{p}_1^{a_1},\ldots,x+\mathfrak{p}_n^{a_n}).$$