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Problem 28 (Characterizing When Factor Rings Are Fields)
Suppose $R$ is a commutative ring with unity and $A$ is an ideal. Prove that the following are equivalent.
- $R/A$ is a field.
- For every $b\in R$ with $b\notin A$, the set $B=\{bc+a\mid c\in R, a\in A\}$ contains the unity $1$ from $R$.
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