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[parent] existence of maximal ideals (Theorem)
EveryRingHasAMaximalIdeal

"existence of maximal ideals" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: Zorn's lemma, axiom of choice, maximal ideal, existence of maximal subgroups, definition of prime ideal by Artin

Other names:  existence of maximal ideals
Keywords:  maximal ideal, commutative ring, identity, axiom of choice

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Cross-references: ZF, axiom of choice, non-associative rings, ring multiplication, associativity, QED, maximal element, belong, ring, without loss of generality, totally ordered, indices, ideals, upper bound, inclusion, partially ordered set, Zorn's lemma, application, proof, zero ideal, theorem, maximal ideal, proper ideal, unital ring
There are 2 references to this entry.

This is version 19 of existence of maximal ideals, born on 2003-09-08, modified 2006-12-25.
Object id is 4713, canonical name is EveryRingHasAMaximalIdeal.
Accessed 5120 times total.

Classification:
AMS MSC: 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
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stronger result by Wkbj79 on 2006-07-30 13:34:01
In my entry $V(I)=\emptyset$ implies $I=R$, I use the result that every proper ideal is contained in a maximal ideal. For this reason, I was tempted to add this result as an entry to PM.

After seeing this object (existence of maximal ideals) though, I realize that it could be adapted to supply a proof for the result that I wanted to add. This can be almost totally accomplished by replacing (0) with I.

I would really like for the result that I discussed here to appear on PM. Should I add it, or should I file an addendum to this object?

Warren
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the theorem is true for non commutative rings also by remag12 on 2004-09-16 08:10:30
The theorem is true for non commutative rings also.
 -- S. A. G.
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