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More lecture notes (#126)

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Patrick Stevens
2020-04-19 20:15:02 +01:00
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parent 485b27e009
commit cab004f6d8
11 changed files with 163 additions and 36 deletions
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7
Rings/Ideals/Maximal/Lemmas.agda
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@@ -80,12 +80,7 @@ MaximalIdeal.isMaximal (quotientFieldImpliesIdealMaximal f) {bigger} biggerIdeal
v = Ideal.isSubset biggerIdeal (invTwice additiveGroup 1R) (Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (Ideal.closedUnderPlus biggerIdeal t u)))
idealMaximalImpliesIdealPrime : ({d : _} → MaximalIdeal i {d}) → PrimeIdeal i
idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i f'
where
f : Field (cosetRing R i)
f = idealMaximalImpliesQuotientField max
f' : IntegralDomain (cosetRing R i)
f' = fieldIsIntDom f (λ p → Field.nontrivial f (Equivalence.symmetric (Setoid.eq (cosetSetoid additiveGroup (Ideal.isSubgroup i))) p))
idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i (fieldIsIntDom (idealMaximalImpliesQuotientField max))
maximalIdealWellDefined : {d : _} {pred2 : A → Set d} (i2 : Ideal R pred2) → ({x : A} → pred x → pred2 x) → ({x : A} → pred2 x → pred x) → {e : _} → MaximalIdeal i {e} → MaximalIdeal i2 {e}
MaximalIdeal.notContained (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) = notContained