nil is a radical property

We must show that the nil property, $\\mathcal{N}$ , is a radical property, that is that it satisfies the following conditions:

1.
The class of $\\mathcal{N}$ -rings is closed under homomorphic images.
2.
Every ring $R$ has a largest $\\mathcal{N}$ -ideal, which contains all other $\\mathcal{N}$ -ideals of $R$ . This ideal is written $\\mathcal{N}(R)$ .
3.
$\\mathcal{N}(R/\\mathcal{N}(R))=0$ .

It is easy to see that the homomorphic image of a nil ring is nil, for if $f\\colon R\\to S$ is a homomorphism and $x^{n}=0$ , then $f(x)^{n}=f(x^{n})=0$ .

The sum of all nil ideals is nil (see proof http://planetmath.org/node/5650here), so this sum is the largest nil ideal in the ring.

Finally, if $N$ is the largest nil ideal in $R$ , and $I$ is an ideal of $R$ containing $N$ such that $I/N$ is nil, then $I$ is also nil (see proof http://planetmath.org/node/5650here). So $I\\subseteq N$ by definition of $N$ . Thus $R/N$ contains no nil ideals.

单词	NilIsARadicalProperty
释义	nil is a radical property We must show that the nil property, $\\mathcal{N}$ , is a radical property, that is that it satisfies the following conditions: 1. The class of $\\mathcal{N}$ -rings is closed under homomorphic images. 2. Every ring $R$ has a largest $\\mathcal{N}$ -ideal, which contains all other $\\mathcal{N}$ -ideals of $R$ . This ideal is written $\\mathcal{N}(R)$ . 3. $\\mathcal{N}(R/\\mathcal{N}(R))=0$ . It is easy to see that the homomorphic image of a nil ring is nil, for if $f\\colon R\\to S$ is a homomorphism and $x^{n}=0$ , then $f(x)^{n}=f(x^{n})=0$ . The sum of all nil ideals is nil (see proof http://planetmath.org/node/5650here), so this sum is the largest nil ideal in the ring. Finally, if $N$ is the largest nil ideal in $R$ , and $I$ is an ideal of $R$ containing $N$ such that $I/N$ is nil, then $I$ is also nil (see proof http://planetmath.org/node/5650here). So $I\\subseteq N$ by definition of $N$ . Thus $R/N$ contains no nil ideals.
随便看	condition for power basis ConditionForUniformConvergenceOfSequenceOfFunctions condition for uniform convergence of sequence of functions ConditionOfOrthogonality condition of orthogonality ConditionOnANearRingToBeARing condition on a near ring to be a ring ConditionsForACollectionOfSubsetsToBeABasisForSomeTopology conditions for a collection of subsets to be a basis for some topology ConductorOfAnEllipticCurve conductor of an elliptic curve ConductorOfAVector conductor of a vector Cone cone cone Cone1 ConeInmathbbR3 cone in ℝ^3 Confluence confluence Confocal confocal ConformallyEquivalent conformally equivalent