Field

Condition for a quotient commutative ring to be a field

Let be a commutative ring and be an ideal. Then the quotient ring is a field iff is a Maximal ideal. ring

Proof

Assume is a field and . Let so that , whence there exists such that . Since ,

whence and therefore , implying .

For the converse, let be maximal. Since is automatically a commutative ring, it remains only to show that is a division ring. Let and

be the ideal generated by . Since is maximal, and in particular . Hence for some and , thus

as required.

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