axioms

[Class 20] Ring Axioms
  1. addition properties - you know all these! (note: can write informally as well as in first-order syntax)
    1. associative $\forall x,y,z[ (x+y)+z = x+(y+z) ]$
    2. commutative $\forall x,y[ x+y = y+x ]$
    3. additive identity $\exists x [ \forall y[ x + y = y ]]$ ← We proved this "x" is unique and decided to call it "0"
    4. additive inverse - $\forall x[ \exists y[ x + y = 0 ] ]$ ← Note:the natural numbers don't have this, so not a ring!
  2. multiplication properties - you know these too
    1. associative $\forall x,y,z[ (x*y)*z = x*(y*z) ]$
    2. commutative $\forall x,y[ x*y = y*x ]$ ← IMPORTANT! not required to be a ring, but is an axiom for integers!
    3. multiplicative identity - we have a constant called 1 such that $\forall x[ 1*x = x ]$
    4. (multiplicative inverse not required ... which is good since almost all integers do not have a multiplicative inverse!)
  3. How multiplication and addition interact
    1. distributivity: $\forall a,b,c[a*(b+c) = a*b + a*c]$ - NOTE: if multiplication is not commutative, you have to define left and right distributivity!

[Class 21] Axioms for total order for rings
Restating the total ordering axioms using the "<" notation we have:
  1. $\forall x[x \nless x]$, i.e. no object is less than itself in the order [note: $a \nless b$ is short-hand for $\neg (a \lt b)$]
  2. $\forall x,y,z[x \lt y \wedge y \lt z \Rightarrow x \lt z]$, i.e. the ordering is transitive
  3. $\forall x,y[x=y \vee x \lt y \vee y \lt x]$, i.e. we have a total order
However, since the objects that are being "ordered" are elements of a ring, we want "$\lt$" to interact with $+$ and $*$ in the way we expect. Technically, we say that the order needs to be compatible with addition and multiplication in the ring. This means:
  1. $\forall x,y,z[x \lt y \Rightarrow x + z \lt y + z]$
  2. $\forall x,y,z[0 \lt z \wedge x \lt y \Rightarrow z*x \lt z*y]$

Definitions

Definition 9: [HW 08] A ring is non-trivial if and only if $\exists x[ x \neq 0 ]$.

Definition 10: [Class 22] A ring is called the integers.

Definition 11: [Class 23] $n$ is even if and only if $\exists k[n=2*k]$.
Note: In full first order logic this would be written as $\forall n[\text{even}(n) \Leftrightarrow \exists k[n=2*k]]$.

Definition 12: [Class 23] $n$ is odd if and only if $\exists k[n=2*k+1]$.
Note: In full first order logic this would be written as $\forall n[\text{odd}(n) \Leftrightarrow \exists k[n=2*k+1]]$.

Theorems

Theorem 0: [Class 19] The additive identity is unique.

Theorem 1: [Class 19] the additive inverse is unique

Theorem 2: [Class 19] $\forall x,y[ x + y = y \Rightarrow x = 0 ]$ , i.e. there are no part-time zeros.

Theorem 3: [Class 19] z*0 = 0 for any z

Theorem 4: [Class 19] $-x = -1*x$.

Theorem 5: [Class 21] (Negation & Order) In a ring with a total order, for any $x$ we have: if $0 \lt x$ then $-x \lt 0$; if $x \lt 0$ then $0 \lt -x$.

Theorem 6: [Class 21] (Trichotemy Law) In a ring with a total order, for any $x$, exactly one of $x \lt 0$, $x = 0$, $0 \lt x$ is true.

Theorem 7: [HW 08] In a non-trivial ring, $0 \neq 1$.

Theorem 8: [HW 08] In a ring that is non-trivial and has a total order, $0 \lt 1$.

Theorem 9: [Class 22] (Gap theorem) For any non-negative integer $x$, $x \lt 1 \Rightarrow x = 0$.
Note: Read this as "the only non-negative integer less than 1 is 0.

Theorem 10: [Class 22] (Gaps everywhere) For any integers $x$ and $k$, if $x \lt k$, then $x = k-1$ or $x \lt k-1$.

Theorem 11: [Class 22] (Generating Positive Integers) Every non-negative integer $k$ is equal to the sum of 0 and $k$ ones - i.e. $0 \underbrace{+ 1 + 1 + \cdots + 1}_{k \text{ copies of } +1}$.

Theorem 12: [Class 24] Zero is not odd.

Theorem 13: [Class 24] No integer is both even and odd.

Theorem 14: [Class 24] Every integer is either even or odd (but, by the previous theorem, not both!).

Theorem 15: [HW 09] (no zero divisors) Over the integers, if $x*y = 0$ then $x = 0$ or $y = 0$.

Theorem 16: [HW 09] (Removing common factors) Over the integers, if $a*b = a*c$ then $a = 0$ or $b = c$.

Christopher W Brown