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∈R. Thus, there exists z∈ℤ with s=zr.
Let t∈R and u∈S. Then u∈R. Since multiplication is commutative
in a cyclic ring, tu=ut. Since t∈R, there exists a∈ℤ with t=ar. Since u∈S, there exists b∈ℤ with u=bs.
Since R is a ring, r2∈R. Thus, there exists k∈ℤ with r2=kr. Since tu=(ar)(bs)=(ar)[b(zr)]=(abz)r2=(abz)(kr)=(abkz)r=(abk)(zr)=(abk)s∈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 |