In mathematics, more specifically ring theory, an ideal, I, of a ring is said to be a nilpotent ideal, if there exists a natural number k such that Ik = 0. By Ik, it is meant the additive subgroup generated by the set of all products of k elements in I. Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0.
The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide—this is exemplified by Levitzky's theorem. The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.
Read more about Nilpotent Ideal: Relation To Nil Ideals, See Also
Famous quotes containing the word ideal:
“Mozart has the classic purity of light and the blue ocean; Beethoven the romantic grandeur which belongs to the storms of air and sea, and while the soul of Mozart seems to dwell on the ethereal peaks of Olympus, that of Beethoven climbs shuddering the storm-beaten sides of a Sinai. Blessed be they both! Each represents a moment of the ideal life, each does us good. Our love is due to both.”
—Henri-Frédéric Amiel (1821–1881)