$V(I)=\\emptyset$ implies $I=R$

Note that most of the notation used here is defined in the entry prime spectrum.

Theorem.

If $R$ is a commutative ring with identity and $I$ is an ideal of $R$ with $V(I)=\\emptyset$ , then $I=R$ .

Proof.

Let $R$ be a commutative ring with identity and $I$ be an ideal of $R$ with $I\eq R$ . Then, by this theorem (http://planetmath.org/EveryRingHasAMaximalIdeal), there exists a maximal ideal $M$ of $R$ containing $I$ . Since $M$ is , then $M$ is a proper prime ideal of $R$ . Thus, $M\\in V(I)$ . The theorem follows.∎

单词	VIemptysetImpliesIR
释义	$V(I)=\\emptyset$ implies $I=R$ Note that most of the notation used here is defined in the entry prime spectrum. Theorem. If $R$ is a commutative ring with identity and $I$ is an ideal of $R$ with $V(I)=\\emptyset$ , then $I=R$ . Proof. Let $R$ be a commutative ring with identity and $I$ be an ideal of $R$ with $I\eq R$ . Then, by this theorem (http://planetmath.org/EveryRingHasAMaximalIdeal), there exists a maximal ideal $M$ of $R$ containing $I$ . Since $M$ is , then $M$ is a proper prime ideal of $R$ . Thus, $M\\in V(I)$ . The theorem follows.∎
随便看	Poénaru (1976) theorem PprimaryComponent p-primary component pqShuffle (p,q) shuffle pqUnshuffle (p,q) unshuffle PracticalNumber practical number practical number PracticalNumber1 PraeclarumTheorema praeclarum theorema PrattCertificate Pratt certificate PrecisionRecall precision recall PrecompactSet precompact set PredecessorsAndSuccesorsInQuivers predecessors and succesors in quivers Predicativism predicativism PredictableProcess predictable process

V⁢(I)=∅ implies I=R

Theorem.

Proof.

$V(I)=\\emptyset$ implies $I=R$