Sums in commutative rings

module commutative-algebra.sums-commutative-rings where
Imports
open import commutative-algebra.commutative-rings

open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.coproduct-types
open import foundation.functions
open import foundation.homotopies
open import foundation.identity-types
open import foundation.unit-type
open import foundation.universe-levels

open import linear-algebra.vectors
open import linear-algebra.vectors-on-commutative-rings

open import ring-theory.sums-rings

open import univalent-combinatorics.coproduct-types
open import univalent-combinatorics.standard-finite-types

Idea

The sum operation extends the binary addition operation on a commutative ring A to any family of elements of A indexed by a standard finite type.

Definition

sum-Commutative-Ring :
  {l : Level} (A : Commutative-Ring l) (n : ) 
  (functional-vec-Commutative-Ring A n)  type-Commutative-Ring A
sum-Commutative-Ring A = sum-Ring (ring-Commutative-Ring A)

Properties

Sums of one and two elements

module _
  {l : Level} (A : Commutative-Ring l)
  where

  sum-one-element-Commutative-Ring :
    (f : functional-vec-Commutative-Ring A 1) 
    sum-Commutative-Ring A 1 f  head-functional-vec 0 f
  sum-one-element-Commutative-Ring =
    sum-one-element-Ring (ring-Commutative-Ring A)

  sum-two-elements-Commutative-Ring :
    (f : functional-vec-Commutative-Ring A 2) 
    sum-Commutative-Ring A 2 f 
    add-Commutative-Ring A (f (zero-Fin 1)) (f (one-Fin 1))
  sum-two-elements-Commutative-Ring =
    sum-two-elements-Ring (ring-Commutative-Ring A)

Sums are homotopy invariant

module _
  {l : Level} (A : Commutative-Ring l)
  where

  htpy-sum-Commutative-Ring :
    (n : ) {f g : functional-vec-Commutative-Ring A n} 
    (f ~ g)  sum-Commutative-Ring A n f  sum-Commutative-Ring A n g
  htpy-sum-Commutative-Ring = htpy-sum-Ring (ring-Commutative-Ring A)

Sums are equal to the zero-th term plus the rest

module _
  {l : Level} (A : Commutative-Ring l)
  where

  cons-sum-Commutative-Ring :
    (n : ) (f : functional-vec-Commutative-Ring A (succ-ℕ n)) 
    {x : type-Commutative-Ring A}  head-functional-vec n f  x 
    sum-Commutative-Ring A (succ-ℕ n) f 
    add-Commutative-Ring A
      ( sum-Commutative-Ring A n (tail-functional-vec n f)) x
  cons-sum-Commutative-Ring = cons-sum-Ring (ring-Commutative-Ring A)

  snoc-sum-Commutative-Ring :
    (n : ) (f : functional-vec-Commutative-Ring A (succ-ℕ n)) 
    {x : type-Commutative-Ring A}  f (zero-Fin n)  x 
    sum-Commutative-Ring A (succ-ℕ n) f 
    add-Commutative-Ring A
      ( x)
      ( sum-Commutative-Ring A n (f  inr-Fin n))
  snoc-sum-Commutative-Ring = snoc-sum-Ring (ring-Commutative-Ring A)

Multiplication distributes over sums

module _
  {l : Level} (A : Commutative-Ring l)
  where

  left-distributive-mul-sum-Commutative-Ring :
    (n : ) (x : type-Commutative-Ring A)
    (f : functional-vec-Commutative-Ring A n) 
    mul-Commutative-Ring A x (sum-Commutative-Ring A n f) 
    sum-Commutative-Ring A n  i  mul-Commutative-Ring A x (f i))
  left-distributive-mul-sum-Commutative-Ring =
    left-distributive-mul-sum-Ring (ring-Commutative-Ring A)

  right-distributive-mul-sum-Commutative-Ring :
    (n : ) (f : functional-vec-Commutative-Ring A n)
    (x : type-Commutative-Ring A) 
    mul-Commutative-Ring A (sum-Commutative-Ring A n f) x 
    sum-Commutative-Ring A n  i  mul-Commutative-Ring A (f i) x)
  right-distributive-mul-sum-Commutative-Ring =
    right-distributive-mul-sum-Ring (ring-Commutative-Ring A)

Interchange law of sums and addition in a commutative ring

module _
  {l : Level} (A : Commutative-Ring l)
  where

  interchange-add-sum-Commutative-Ring :
    (n : ) (f g : functional-vec-Commutative-Ring A n) 
    add-Commutative-Ring A
      ( sum-Commutative-Ring A n f)
      ( sum-Commutative-Ring A n g) 
    sum-Commutative-Ring A n
      ( add-functional-vec-Commutative-Ring A n f g)
  interchange-add-sum-Commutative-Ring =
    interchange-add-sum-Ring (ring-Commutative-Ring A)

Extending a sum of elements in a commutative ring

module _
  {l : Level} (A : Commutative-Ring l)
  where

  extend-sum-Commutative-Ring :
    (n : ) (f : functional-vec-Commutative-Ring A n) 
    sum-Commutative-Ring A
      ( succ-ℕ n)
      ( cons-functional-vec-Commutative-Ring A n (zero-Commutative-Ring A) f) 
    sum-Commutative-Ring A n f
  extend-sum-Commutative-Ring = extend-sum-Ring (ring-Commutative-Ring A)

Shifting a sum of elements in a commutative ring

module _
  {l : Level} (A : Commutative-Ring l)
  where

  shift-sum-Commutative-Ring :
    (n : ) (f : functional-vec-Commutative-Ring A n) 
    sum-Commutative-Ring A
      ( succ-ℕ n)
      ( snoc-functional-vec-Commutative-Ring A n f
        ( zero-Commutative-Ring A)) 
    sum-Commutative-Ring A n f
  shift-sum-Commutative-Ring = shift-sum-Ring (ring-Commutative-Ring A)

Splitting sums

split-sum-Commutative-Ring :
  {l : Level} (A : Commutative-Ring l)
  (n m : ) (f : functional-vec-Commutative-Ring A (n +ℕ m)) 
  sum-Commutative-Ring A (n +ℕ m) f 
  add-Commutative-Ring A
    ( sum-Commutative-Ring A n (f  inl-coprod-Fin n m))
    ( sum-Commutative-Ring A m (f  inr-coprod-Fin n m))
split-sum-Commutative-Ring A n zero-ℕ f =
  inv (right-unit-law-add-Commutative-Ring A (sum-Commutative-Ring A n f))
split-sum-Commutative-Ring A n (succ-ℕ m) f =
  ( ap
    ( add-Commutative-Ring' A (f(inr star)))
    ( split-sum-Commutative-Ring A n m (f  inl))) 
  ( associative-add-Commutative-Ring A _ _ _)

A sum of zeroes is zero

module _
  {l : Level} (A : Commutative-Ring l)
  where

  sum-zero-Commutative-Ring :
    (n : ) 
    sum-Commutative-Ring A n
      ( zero-functional-vec-Commutative-Ring A n) 
    zero-Commutative-Ring A
  sum-zero-Commutative-Ring = sum-zero-Ring (ring-Commutative-Ring A)