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Commutative algebra

Trivial ring

A commutative ring $R$ is trivial if $0 = 1$.

Discrete rings

A commutative ring $R$ is discrete if for all elements $x:R$ and $y:R$, either $x = y$, or $x = y$ implies $0 = 1$.

Regular elements

Given a commutative ring $R$, a term $e:R$ is left regular if for all $a:R$ and $b:R$, $e \cdot a = e \cdot b$ implies $a = b$.

A term $e:R$ is right regular if for all $a:R$ and $b:R$, $a \cdot e = b \cdot e$ implies $a = b$.

An term $e:R$ is regular if it is both left regular and right regular.

The multiplicative monoid of regular elements in $R$ is the submonoid of all regular elements in $R$

Invertible elements

Given a commutative ring $R$, a term $e:R$ is left invertible if the fiber of right multiplication by $e$ at $1$ is inhabited.

A term $e:R$ is right invertible if the fiber of left multiplication by $e$ at $1$ is inhabited.

An term $e:R$ is invertible if it is both left invertible and right invertible.

The multiplicative group of invertible elements in $R$ is the subgroup of all invertible elements in $R$

Non-regular and non-invertible elements

An element $x:R$ is non-regular if $x$ being regular implies that $0 = 1$

An element $x:R$ is non-invertible if $x$ being invertible implies that $0 = 1$

Zero is always a non-regular and non-invertible element of $R$. By definition of non-regular and non-invertible, if $0$ is regular or invertible, then the ring $R$ is trivial.

A commutative ring is integral if every non-regular element is equal to zero. A commutative ring is a field if every non-invertible element is equal to zero.

References

• Frank Quinn, Proof Projects for Teachers (pdf)

• Henri Lombardi, Claude Quitté, Commutative algebra: Constructive methods (Finite projective modules) (arXiv:1605.04832)

Real numbers

Pointed abelian groups

A pointed abelian group is a set $A$ with an element $1:A$ called one and a binary operation $(-)-(-):A \times A \to A$ called subtraction such that

• for all elements $a:A$, $a - a = 1 - 1$

• for all elements $a:A$, $(1 - 1) - ((1 - 1) - a) = a$

• for all elements $a:A$ and $b:A$, $a - ((1 - 1) - b) = b - ((1 - 1) - a)$

• for all elements $a:A$, $b:A$, and $c:A$, $a - (b - c) = (a - ((1 - 1) - c)) - b$

Pointed halving groups

A pointed halving group is a pointed abelian group G with a function $(-)/2:G \to G$ called halving or dividing by two such that for all elements $g:G$, $g/2 = g - g/2$.

Totally ordered pointed halving groups

A pointed halving group $R$ is a totally ordered pointed halving group if it comes with a function $\max:R \times R \to R$ such that

• for all elements $a:R$, $\max(a, a) = a$

• for all elements $a:R$ and $b:R$, $\max(a, b) = \max(b, a)$

• for all elements $a:R$, $b:R$, and $c:R$, $\max(a, \max(b, c)) = \max(\max(a, b), c)$

• for all elements $a:R$ and $b:R$, $\max(a, b) = a$ or $\max(a, b) = b$

• for all elements $a:R$ and $b:R$, $\max(a, b) = b$ implies that for all elements $c:R$, $\max(a - c, b - c) = b - c$

Strictly ordered pointed halving groups

A totally ordered pointed halving group $R$ is a strictly ordered pointed abelian group if it comes with a type family $\lt$ such that

• for all elements $a:R$ and $b:R$, $a \lt b$ is a proposition
• for all elements $a:R$, $a \lt a$ is false
• for all elements $a:R$, $b:R$, and $c:R$, if $a \lt c$, then $a \lt b$ or $b \lt c$
• for all elements $a:R$ and $b:R$, if $a \lt b$ is false and $b \lt a$ is false, then $a = b$
• for all elements $a:R$ and $b:R$, if $a \lt b$, then $b \lt a$ is false.
• $1 - 1 \lt 1$
• for all elements $a:R$ and $b:R$, if $1 - 1 \lt a$ and $1 - 1 \lt b$, then $(1 - 1) \lt a - ((1 - 1) - b)$

The homotopy initial strictly ordered pointed halving group is the dyadic rational numbers $\mathbb{D}$.

Archimedean ordered pointed halving groups

A strictly ordered pointed halving group $A$ is an Archimedean ordered pointed halving group if for all elements $a:A$ and $b:A$, if $a \lt b$, then there merely exists a dyadic rational number $d:\mathbb{D}$ such that $a \lt h(d)$ and $h(d) \lt b$.

Sequentially Cauchy complete Archimedean ordered pointed halving groups

Let $A$ be an Archimedean ordered pointed halving group and let

be the positive elements in $A$.

A sequence in $A$ is a function $x:\mathbb{N} \to A$.

A sequence $x:\mathbb{N} \to A$ is a Cauchy sequence if for all positive elements $\epsilon:A_{+}$, there merely exists a natural number $N:\mathbb{N}$ such that for all natural numbers $i:\mathbb{N}$ and $j:\mathbb{N}$ such that $N \leq i$ and $N \leq j$, $\max(x_i - x_j, x_j - x_i) \lt \epsilon$.

An element $l:A$ is said to be a limit of the sequence $x:\mathbb{N} \to A$ if for all positive elements $\epsilon:A_{+}$, there merely exists a natural number $N:\mathbb{N}$ such that for all natural numbers $i:\mathbb{N}$ such that $N \leq i$, $\max(x_i - l, l - x_i) \lt \epsilon$

$A$ is sequentially Cauchy complete if every Cauchy sequence in $A$ merely has a limit.