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Homeproof that every subring of a cyclic ring is a cyclic ring

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# proof that every subring of a cyclic ring is a cyclic ring

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

###### Proof.

Let $R$ be a cyclic ring and $S$ be a subring of $R$. Then the additive group of $S$ is a subgroup of the additive group of $R$. By definition of cyclic group, the additive group of $R$ is cyclic. Thus, the additive group of $S$ is cyclic. It follows that $S$ is a cyclic ring. ∎

Related:

ProofThatAllSubgroupsOfACyclicGroupAreCyclic, ProofThatEverySubringOfACyclicRingIsAnIdeal

Major Section:

Reference

Type of Math Object:

Proof

Parent:

## Mathematics Subject Classification

13A99*no label found*16U99

*no label found*

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