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I'm reading William Stein's: Elementary Number Theory: Primes, Congruences, and Secrets. And I found this definition.

It employs the concept of reduction map and lift, but it seems to be very similar to function and inverse image (not inverse function). As the book is not so clear about the definition of these concepts, are there any benefits of using them?

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1 Answer

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Of course, the reduction map is itself a function (a surjective one) and you're right, a lift for an element on $Z/nZ$ is just an element of its inverse image.

However, there are benefits in using those terms since we refer to them a lot and it's easier to say "take a lift of $a+nZ$" instead of "take an element of $f^{-1}(a+nZ)$" (however, the word lift is not used very often in the literature, but reduction is).

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