Nil Ideal - Commutative Rings

Commutative Rings

In a commutative ring, the set of all nilpotent elements forms an ideal known as the nilradical of the ring. Therefore, an ideal of a commutative ring is nil if, and only if, it is a subset of the nilradical; that is, the nilradical is the ideal maximal with respect to the property that each of its elements is nilpotent.

In commutative rings, the nil ideals are more well-understood compared to the case of noncommutative rings. This is primarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent. For instance, if a is a nilpotent element of a commutative ring R, a·R is an ideal that is in fact nil. This is because any element of the principal ideal generated by a is of the form a·r for r in R, and if an = 0, (a·r)n = an·rn = 0. It is not in general true however, that a·R is a nil (one-sided) ideal in a noncommutative ring, even if a is nilpotent.