Nilpotent ideal

A one- or two-sided ideal $M$ in a ring or semi-group with zero such that $M^n=\{0\}$ for some natural number $n$, that is, the product of any $n$ elements of $M$ vanishes. For example, in the residue class ring $\mathbf Z/p^n\mathbf Z$ modulo $p^n$, where $p$ is a prime number, every ideal except the ring itself is nilpotent. In the group ring $\mathbf F_p[G]$ of a finite $p$-group $G$ over the field with $p$ elements the ideal generated by the elements of the form $\sigma-1$, $\sigma\in G$, is nilpotent. In the ring of upper-triangular matrices over a field the matrices with 0's along the main diagonal form a nilpotent ideal.

Every element of a nilpotent ideal is nilpotent. Every nilpotent ideal is also a nil ideal and is contained in the Jacobson radical of the ring. In Artinian rings the Jacobson radical is nilpotent, and the concepts of a nilpotent ideal and a nil ideal coincide. The latter property also holds in a Noetherian ring. In a left (or right) Noetherian ring every left (right) nil ideal is nilpotent.

All nilpotent ideals of a commutative ring are contained in the nil radical, which, in general, need not be a nilpotent but only a nil ideal. A simple example of this situation is the direct sum of the rings $\mathbf Z/p^n\mathbf Z$ for all natural numbers $n$. In a commutative ring every nilpotent element $a$ is contained in some nilpotent ideal, for example, in the principal ideal generated by $a$. In a non-commutative ring there may by nilpotent elements that are not contained in any nilpotent ideal (nor even in a nil ideal). For example, in the general matrix ring over a field there are nilpotent elements; in particular, the nilpotent matrices mentioned above, in which the only non-zero elements stand above the main diagonal, but since the ring is simple, it has no non-zero nilpotent ideals.

In a finite-dimensional Lie algebra $G$ there is maximal nilpotent ideal, which consists of the elements $x\in G$ for which the endomorphism $y\to[x,y]$ for $y\in G$ is nilpotent.

References

[1]	S. Lang, "Algebra" , Addison-Wesley (1974)
[2]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[3]	C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[4]	I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)
[5]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)