Take the 2-minute tour ×  Self-explanatory title really! A
student today asked me why they were called integral domains --
and I realised that the word "integral" seems to be being used
in a way totally unlike any other way I hear it used in
mathematics. The student suggested that "integral" was used
because the integers were an example -- but I didn't buy this
because by that logic they could have been called "rational
domains" or "real domains".  Any ideas anyone?  share|improve
this question Here's a quote from Earliest known uses..., though
it's almost surely not the earliest: `INTEGRAL DOMAIN is found
in 1911 in Monographs on Topics of Modern Mathematics: Relevant
to the Elementary Field by numerous writers: "Similarly, the
integral domain [R1, R2, R3, ...] may be defined by replacing
the expressions rational functions and rational coefficients in
the preceding definition by integral functions and integral
coefficients respectively."' –  Bill Dubuque Jun 17 '11 at 18:35
I don't know if it is the origin,