cyclic rings of behavior one

Theorem.

A cyclic ring has a multiplicative identity if and only if it has behavior one.

Proof.

For a proof that a cyclic ring with a multiplicative identity has behavior one, see this theorem (http://planetmath.org/MultiplicativeIdentityOfACyclicRingMustBeAGenerator).

Let $R$ be a cyclic ring with behavior one. Let $r$ be a generator (http://planetmath.org/Generator) of the additive group of $R$ such that $r^{2}=r$ . Let $s\\in R$ . Then there exists $a\\in R$ with $s=ar$ . Since $rs=r(ar)=ar^{2}=ar=s$ and multiplication in cyclic rings is commutative, then $r$ is a multiplicative identity.∎

单词	CyclicRingsOfBehaviorOne
释义	cyclic rings of behavior one Theorem. A cyclic ring has a multiplicative identity if and only if it has behavior one. Proof. For a proof that a cyclic ring with a multiplicative identity has behavior one, see this theorem (http://planetmath.org/MultiplicativeIdentityOfACyclicRingMustBeAGenerator). Let $R$ be a cyclic ring with behavior one. Let $r$ be a generator (http://planetmath.org/Generator) of the additive group of $R$ such that $r^{2}=r$ . Let $s\\in R$ . Then there exists $a\\in R$ with $s=ar$ . Since $rs=r(ar)=ar^{2}=ar=s$ and multiplication in cyclic rings is commutative, then $r$ is a multiplicative identity.∎
随便看	Square of a generic sum of elements SquareOfSum square of sum SquareRoot square root SquareRootOf2 square root of 2 SquareRootOf3 square root of 3 SquareRootOf3567And29 square root of 3, 5, 6, 7 and 29 SquareRootOf5 square root of 5 SquareRootOfPolynomial square root of polynomial SquareRootOfPositiveDefiniteMatrix square root of positive definite matrix SquareRootOfSquareRootBinomial square root of square root binomial SquareRootsOfRationals square roots of rationals SquaringConditionForSquareRootInequality squaring condition for square root inequality SqueezeRule squeeze rule