## You are here

Homeproof that every subring of a cyclic ring is an ideal

## Primary tabs

# 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 generator of the additive group of $R$ and $s$ be a generator of the additive group of $S$. Then $s\in R$. Thus, there exists $z\in\mathbb{Z}$ with $s=zr$.

Let $t\in R$ and $u\in S$. Then $u\in R$. Since multiplication is commutative in a cyclic ring, $tu=ut$. Since $t\in R$, there exists $a\in{\mathbb{Z}}$ with $t=ar$. Since $u\in S$, there exists $b\in{\mathbb{Z}}$ with $u=bs$.

# References

- 1
Buck, Warren.
*Cyclic 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.

Related:

ProofThatEverySubringOfACyclicRingIsACyclicRing

Major Section:

Reference

Type of Math Object:

Proof

Parent:

## Mathematics Subject Classification

13A99*no label found*16U99

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections