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In Bourbaki's "Commutative Algebra", p. 386, the valuation of a ring (and later a field) is not required to be surjective. The same is true for the definition that Hartshorne uses in his "Algebraic Geometry". p. 39. On the contrary, Atiyah-MacDonald define a discrete valuation in p. 94 to be a surjective mapping onto $\mathbb{Z}$.

How does this discrepancy affect the existence of a uniformizer (local parameter in algebraic geometry context) for discrete valuation rings?

In Bourbaki e.g. p. 392, the reference to the uniformizer is almost axiomatic; i can not see any proof of existence. On the other hand, the existence in Atiyah-MacDonald follows immediately from surjectivity.

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    The image is infinite cyclic, so you may assume it is surjective.2012-10-23

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In general, the value group is only assumed to be discrete. Then there is an element $\pi$ with smallest positive valuation, and indeed the value group is generated by $v(\pi)$. So one may normalize it by defining $v'(x) := v(x)/v(\pi)$ (the valuation ring doesn't change), so that $v'(\pi)=1$ and the value group becomes $\mathbb{Z}$.