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user 93853
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Nonequivalent definitions in Mathematics
These defintions coincide iff $R$ is a PID, hence for higher-dimensional UFD's (like $k[x,y]$) we get two different definitions of the same notion. … Also note the slightly different definitions of a projective morphism in EGA and Hartshorne. To be fair Hartshorne adresses the difference after his definition. …