In page 3 of Lei Fu's "Etale Cohomology Theory", there is a notion called discrete valuation ring as follows:
But about discrete valuation, I have found two different definitions.
One is from Singh's "Basic Commutative Algebra", which is as follows:
The other one is from Atiyah's "An Introduction to Commutative Algebra", which is as follows:
For Lei Fu's "Etale Cohomology Theory", which definition should I choose?
The only difference is that Atiyah requires a discrete valuation to be surjective, but any non-surjective discrete valuation $v$ (as per Singh) can be made surjective: its image is $m\Bbb Z$ for some natural $m$, so we can define a new valuation $v'$ by $v'= \frac{1}{m}v$.