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^{} (http://planetmath.org/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 $u=bs$.
Since $R$ is a ring, ${r}^{2}\in R$. Thus, there exists $k\in \mathbb{Z}$ with ${r}^{2}=kr$. Since $tu=(ar)(bs)=(ar)[b(zr)]=(abz){r}^{2}=(abz)(kr)=(abkz)r=(abk)(zr)=(abk)s\in S$, 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 |
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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 |