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Problem 17 (Properties Needed For A Factor Group To Be A Factor Ring)
Suppose $R$ is a ring and $A$ is a subring of $R$. Because $A$ is a subgroup of $R$, we know that $R/A$ is an Abelian group, as factor groups of Abelian groups are Abelian. Recall that $$R/A = \{r+A \mid r\in R \},$$ the collection of cosets of $A$.
- What would it take for $R/A$ to be a ring? What properties must we satisfy? Write out the needed properties.
- Suppose that $R/A$ is a ring. What additional facts about the cosets must hold to guarantee that $R/A$ is an integral domain?
- If $R/A$ is an integral domain, what would we need to guarantee that $R/A$ is a field?
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