I have been doing problems in Atiyah & MacDonald's Introduction to Commutative Algebra, and in problem 1.6 it asks to assume the existence of an idempotent element in an ideal whenever the ideal does not lie inside the nilradical. I could have sworn the definition of idempotent was as in the title, but after some investigation, idempotence seems to be reserved for the case where $m=2$.
This sort of blunder could have set me astray, as for example, the hypothesis in problem 1.6 seems to become considerably weaker. Luckily the weakened hypothesis did the job just fine for that problem, but I'd like to prevent future trouble by having a name for the general $m$.
So what is it? I have seen this property before, so I imagine it has been named somewhere, but I cannot seem to find one.