I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following

**General Chinese Remainder Theorem.** For integer numbers $a_1,\dots,a_n$ and positive integers $b_1,\dots,b_n$ the intersection $\bigcap_{i=1}^n(a_i+b_i\mathbb Z)$ is not empty if and only if $a_i-a_j\in d_{i,j}\mathbb Z$ for any $1\le i<j\le n$, where $d_{ij}$ denotes the largest common divisor of the numbers $b_i,b_j$.

Looking at the internet, I found this paper in which the General Chinese Remainder Theorem is formulated as an exercise and another paper in which this theorem is proved.

But I am suspecting that such General Chinese Remainder Theorem should be proved in some standard (undergraduate) textbook in Number Theory. I need it for a proper reference. Please help!

I understand that this is rather a reference request and not a problem of research level. In case of downvotes I will delete it as soon as will get a proper answer from experts.

generalChinese Remainder Theorem. ThegeneralCRT is stated for an arbitrary commutative ring and coprime ideals (and your version directly follows from it), hence you should be able to find it in any book on general abstract algebra. Off top of my head, there is a short proof in the first chapter in Atiyah and MacDonald's Commutative Algebra (which I guess is a little more specialized book), but really any general book on abstract algebra should do. $\endgroup$ – M.G. Nov 21 '17 at 11:01