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. #m/thm/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.

As a corollary, a ring is a field iff it has no nontrivial proper ideals.

#state/tidy | #lang/en | #SemBr