Dear Randy,
"When an expression raised to the square or any higher power
vanishes, it may be called _nilpotent_; but when, raised to
a square or higher power, it gives itself as the result, it
may be called _idempotent_.
The defining equation of nilpotent and idempotent expressions
are respectively A^n = 0, and A^n = A; but with reference to
idempotent expressions, it will always be assumed that they
are of the form
A^2 = A ,
unless it be otherwise distinctly stated."
Excerpted from "Linear Associative Algebra", a memoir read by Benjamin
Peirce before the National Academy of Sciences in Washington, 1870.
I thought it might be worth recalling also that Benjamin opens this
famous memoir with the following colourful definition:
"Mathematics is the science which draws necessary conclusions".
With best regards,
Julio Gonzalez Cabillon