Every semisimple ring is also Artinian, and has no nilpotent ideals.
2.
Every nilpotent ideal is nil, but the converse is not true in general.
3.
An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring.
4.
The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings.
5.
The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.
6.
Every such "'M "'as defined above is a semisimple algebra, i . e . it contains no nilpotent ideals.
7.
*PM : properties of nil and nilpotent ideals, id = 5650-- WP guess : properties of nil and nilpotent ideals-- Status:
8.
*PM : properties of nil and nilpotent ideals, id = 5650-- WP guess : properties of nil and nilpotent ideals-- Status:
9.
If a finite-dimensional Lie algebra is nilpotent, then the Killing form is identically zero ( and more generally the Killing form vanishes on any nilpotent ideal ).
10.
Since the lower nilradical contains all nilpotent ideals, it also contains " N ", and so " N " is equal to the lower nilradical.
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