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Consider a commutative ring R with a prime ideal P. Is there a terminology or name for the field of fractions of the R/P in the literature? I suspect that "function field of P" is not a correct term.

My second question is there any standard (or at least widely used) notation for this field in the literature?

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    Residue field is a pretty common term for this. – 2017-02-25
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    You mean $P$ is a maximal ideal so that $R/P$ is a field (when $R$ is a Dedekind domain every prime ideals are maximal) – 2017-02-25

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This is usually called the "residue field" at $P$, and is often denoted $k(P)$. Both the term and the notation are also used more generally for the corresponding notion when $P$ is a point in a scheme (or more generally a locally ringed space).

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    What is k in k(P)? An abbreviation for "Körper"? Is the notation k(P) also used in the situation where R is not necessarily a k-algebra? If this is the case, I think this completely answers my question. – 2017-02-25
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    Yeah, $k$ here is just used in the sense of "field". It isn't referring to any particular field that $R$ is an algebra over. – 2017-02-25