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Congruence relations on rings and semirings (#494)
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| # Congruence relations on rings | ||
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| ```agda | ||
| module ring-theory.congruence-relations-rings where | ||
| ``` | ||
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||
| <details><summary>Imports</summary> | ||
|
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| ```agda | ||
| open import foundation.binary-relations | ||
| open import foundation.dependent-pair-types | ||
| open import foundation.equivalence-relations | ||
| open import foundation.equivalences | ||
| open import foundation.propositions | ||
| open import foundation.universe-levels | ||
| open import group-theory.congruence-relations-abelian-groups | ||
| open import group-theory.congruence-relations-monoids | ||
| open import ring-theory.congruence-relations-semirings | ||
| open import ring-theory.rings | ||
| ``` | ||
|
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| </details> | ||
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| ## Idea | ||
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| A congruence relation on a ring `R` is a congruence relation on the | ||
| underlying semiring of `R`. | ||
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| ## Definition | ||
|
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| ```agda | ||
| module _ | ||
| {l1 : Level} (R : Ring l1) | ||
| where | ||
| is-congruence-Ring : | ||
| {l2 : Level} → congruence-Ab l2 (ab-Ring R) → UU (l1 ⊔ l2) | ||
| is-congruence-Ring = is-congruence-Semiring (semiring-Ring R) | ||
| is-congruence-eq-rel-Ring : | ||
| {l2 : Level} (S : Eq-Rel l2 (type-Ring R)) → UU (l1 ⊔ l2) | ||
| is-congruence-eq-rel-Ring S = | ||
| is-congruence-eq-rel-Semiring (semiring-Ring R) S | ||
| congruence-Ring : | ||
| {l1 : Level} (l2 : Level) (R : Ring l1) → UU (l1 ⊔ lsuc l2) | ||
| congruence-Ring l2 R = congruence-Semiring l2 (semiring-Ring R) | ||
| module _ | ||
| {l1 l2 : Level} (R : Ring l1) (S : congruence-Ring l2 R) | ||
| where | ||
| congruence-ab-congruence-Ring : congruence-Ab l2 (ab-Ring R) | ||
| congruence-ab-congruence-Ring = | ||
| congruence-additive-monoid-congruence-Semiring (semiring-Ring R) S | ||
| eq-rel-congruence-Ring : Eq-Rel l2 (type-Ring R) | ||
| eq-rel-congruence-Ring = | ||
| eq-rel-congruence-Semiring (semiring-Ring R) S | ||
| prop-congruence-Ring : Rel-Prop l2 (type-Ring R) | ||
| prop-congruence-Ring = prop-congruence-Semiring (semiring-Ring R) S | ||
| sim-congruence-Ring : (x y : type-Ring R) → UU l2 | ||
| sim-congruence-Ring = sim-congruence-Semiring (semiring-Ring R) S | ||
| is-prop-sim-congruence-Ring : | ||
| (x y : type-Ring R) → is-prop (sim-congruence-Ring x y) | ||
| is-prop-sim-congruence-Ring = | ||
| is-prop-sim-congruence-Semiring (semiring-Ring R) S | ||
| refl-congruence-Ring : | ||
| is-reflexive-Rel-Prop prop-congruence-Ring | ||
| refl-congruence-Ring = refl-congruence-Semiring (semiring-Ring R) S | ||
| symm-congruence-Ring : | ||
| is-symmetric-Rel-Prop prop-congruence-Ring | ||
| symm-congruence-Ring = symm-congruence-Semiring (semiring-Ring R) S | ||
| equiv-symm-congruence-Ring : | ||
| (x y : type-Ring R) → | ||
| sim-congruence-Ring x y ≃ sim-congruence-Ring y x | ||
| equiv-symm-congruence-Ring = | ||
| equiv-symm-congruence-Semiring (semiring-Ring R) S | ||
| trans-congruence-Ring : | ||
| is-transitive-Rel-Prop prop-congruence-Ring | ||
| trans-congruence-Ring = | ||
| trans-congruence-Semiring (semiring-Ring R) S | ||
| add-congruence-Ring : | ||
| is-congruence-Ab (ab-Ring R) eq-rel-congruence-Ring | ||
| add-congruence-Ring = add-congruence-Semiring (semiring-Ring R) S | ||
| left-add-congruence-Ring : | ||
| (x : type-Ring R) {y z : type-Ring R} → | ||
| sim-congruence-Ring y z → | ||
| sim-congruence-Ring (add-Ring R x y) (add-Ring R x z) | ||
| left-add-congruence-Ring = | ||
| left-add-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| right-add-congruence-Ring : | ||
| {x y : type-Ring R} → sim-congruence-Ring x y → | ||
| (z : type-Ring R) → | ||
| sim-congruence-Ring (add-Ring R x z) (add-Ring R y z) | ||
| right-add-congruence-Ring = | ||
| right-add-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| sim-right-subtraction-zero-congruence-Ring : (x y : type-Ring R) → UU l2 | ||
| sim-right-subtraction-zero-congruence-Ring = | ||
| sim-right-subtraction-zero-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| map-sim-right-subtraction-zero-congruence-Ring : | ||
| {x y : type-Ring R} → sim-congruence-Ring x y → | ||
| sim-right-subtraction-zero-congruence-Ring x y | ||
| map-sim-right-subtraction-zero-congruence-Ring = | ||
| map-sim-right-subtraction-zero-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| map-inv-sim-right-subtraction-zero-congruence-Ring : | ||
| {x y : type-Ring R} → | ||
| sim-right-subtraction-zero-congruence-Ring x y → sim-congruence-Ring x y | ||
| map-inv-sim-right-subtraction-zero-congruence-Ring = | ||
| map-inv-sim-right-subtraction-zero-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| sim-left-subtraction-zero-congruence-Ring : (x y : type-Ring R) → UU l2 | ||
| sim-left-subtraction-zero-congruence-Ring = | ||
| sim-left-subtraction-zero-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| map-sim-left-subtraction-zero-congruence-Ring : | ||
| {x y : type-Ring R} → sim-congruence-Ring x y → | ||
| sim-left-subtraction-zero-congruence-Ring x y | ||
| map-sim-left-subtraction-zero-congruence-Ring = | ||
| map-sim-left-subtraction-zero-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| map-inv-sim-left-subtraction-zero-congruence-Ring : | ||
| {x y : type-Ring R} → sim-left-subtraction-zero-congruence-Ring x y → | ||
| sim-congruence-Ring x y | ||
| map-inv-sim-left-subtraction-zero-congruence-Ring = | ||
| map-inv-sim-left-subtraction-zero-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| neg-congruence-Ring : | ||
| {x y : type-Ring R} → sim-congruence-Ring x y → | ||
| sim-congruence-Ring (neg-Ring R x) (neg-Ring R y) | ||
| neg-congruence-Ring = | ||
| neg-congruence-Ab | ||
| ( ab-Ring R) | ||
| ( congruence-ab-congruence-Ring) | ||
| mul-congruence-Ring : | ||
| is-congruence-Monoid | ||
| ( multiplicative-monoid-Ring R) | ||
| ( eq-rel-congruence-Ring) | ||
| mul-congruence-Ring = pr2 S | ||
| construct-congruence-Ring : | ||
| {l1 l2 : Level} (R : Ring l1) → | ||
| (S : Eq-Rel l2 (type-Ring R)) → | ||
| is-congruence-Ab (ab-Ring R) S → | ||
| is-congruence-Monoid (multiplicative-monoid-Ring R) S → | ||
| congruence-Ring l2 R | ||
| pr1 (pr1 (construct-congruence-Ring R S H K)) = S | ||
| pr2 (pr1 (construct-congruence-Ring R S H K)) = H | ||
| pr2 (construct-congruence-Ring R S H K) = K | ||
| ``` |
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