proof that every subring of a cyclic ring is an ideal


The following is a proof that every subring of a cyclic ring is an ideal.

Proof.

Let R be a cyclic ring and S be a subring of R. Then R and S are both cyclic rings. Let r be a generatorPlanetmathPlanetmathPlanetmath (http://planetmath.org/Generator) of the additive groupMathworldPlanetmath of R and s be a generator of the additive group of S. Then sR. Thus, there exists z with s=zr.

Let tR and uS. Then uR. Since multiplicationPlanetmathPlanetmath is commutativePlanetmathPlanetmathPlanetmath in a cyclic ring, tu=ut. Since tR, there exists a with t=ar. Since uS, there exists b with u=bs.

Since R is a ring, r2R. Thus, there exists k with r2=kr. Since tu=(ar)(bs)=(ar)[b(zr)]=(abz)r2=(abz)(kr)=(abkz)r=(abk)(zr)=(abk)sS, it follows that S is an ideal of R. ∎

References

  • 1 Buck, Warren. http://planetmath.org/?op=getobj&from=papers&id=336Cyclic Rings. Charleston, IL: Eastern Illinois University, 2004.
  • 2 Maurer, I. Gy. and Vincze, J. “Despre Inele Ciclece.” Studia Universitatis Babeş-Bolyai. Series Mathematica-Physica, vol. 9 #1. Cluj, Romania: Universitatea Babeş-Bolyai, 1964, pp. 25-27.
Title proof that every subring of a cyclic ring is an ideal
Canonical name ProofThatEverySubringOfACyclicRingIsAnIdeal
Date of creation 2013-03-22 13:30:52
Last modified on 2013-03-22 13:30:52
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 9
Author Wkbj79 (1863)
Entry type Proof
Classification msc 13A99
Classification msc 16U99
Related topic ProofThatEverySubringOfACyclicRingIsACyclicRing