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Vipul https://commalg.subwiki.org/w/index.php?title=Maximal_implies_prime&diff=736&oldid=prev 2008-01-24T14:54:05Z

<p>New page: {{curing-ideal property implication}} ==Statement== Any <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> in a <a href="/wiki/Commutative_unital_ring" title="Commutative unital ring">commutative unital ring</a> is a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>. ==Definitions used== ===Maximal ideal=== {{further|[[M...</p> <p><b>New page</b></p><div>{{curing-ideal property implication}}<br /> <br /> ==Statement==<br /> <br /> Any [[maximal ideal]] in a [[commutative unital ring]] is a [[prime ideal]].<br /> <br /> ==Definitions used==<br /> <br /> ===Maximal ideal===<br /> <br /> {{further|[[Maximal ideal]]}}<br /> <br /> ===Prime ideal===<br /> <br /> {{further|[[Prime ideal]]}}<br /> <br /> ==Proof==<br /> <br /> ===As quotient-determined properties===<br /> <br /> One proof uses the characterization of maximal and prime ideals in terms of their quotients, namely:<br /> <br /> * An ideal is maximal iff the quotient ring is a [[field]]<br /> * An ideal is prime iff the quotient ring is an [[integral domain]]<br /> <br /> We know that the [[property of commutative unital rings]] of being a [[field]] is stronger than the property of being an [[integral domain]], hence any maximal ideal is prime.<br /> <br /> ===Hands-on proof===<br /> <br /> {{fillin}}</div>

Vipul