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In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent. [1] [2]The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil.
A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
A video game walkthrough is a guide aimed towards improving a player's skill within a particular video game and often designed to assist players in completing either an entire video game or specific elements. Walkthroughs may alternatively be set up as a playthrough, where players record themselves playing through a game and upload or live ...
If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.
The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring R/I is reduced if and only if I is a radical ...
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical n i l ( g ) {\displaystyle {\mathfrak {nil}}({\mathfrak {g}})} of a finite-dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} is its maximal nilpotent ideal , which exists because the sum of any two nilpotent ideals is nilpotent.
Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. Nilradical may refer to: Nilradical of a ring; Nilradical of a Lie algebra ...
Volition was an American video game developer located in Champaign, Illinois.It was founded in 1993 by programmers Mike Kulas and Matt Toschlog as Parallax Software. The company grew to eight employees while developing its first game, the first-person spaceship shooter Descent (1995), which was released to widespread acclaim.