Ring


Definition 1

ProofWiki

Definition 2

Addition is associative:
\(\forall x,y,z \in S : x + y + z = x + (y + z)\)
Addition is commutative:
\(\forall x,y \in S : x + y = y + x\)
Zero is a right identity element of addition:
\(\forall x \in S : x + 0 = x\)
Zero is a left identity element of addition:
\(\forall x \in S : 0 + x = x\)
Every element has a right additive inverse:
\(\forall x \in S : \exists x' \in S : x + x' = 0\)
Every element has a left additive inverse:
\(\forall x \in S : \exists x' \in S : x' + x = 0\)
Multiplication is associative:
\(\forall x,y,z \in S : xyz = x(yz)\)
Multiplication is left-distributive over addition:
\(\forall x,y,z \in S : x(y + z) = xy + xz\)
Multiplication is right-distributive over addition:
\(\forall x,y,z \in S : (y + z)x = yx + zx\)

Axioms

    add_assoc: ?a + ?b + ?c = ?a + (?b + ?c)

  add_commute: ?a + ?b = ?b + ?a

  add_0_right: ?a + 0 = ?a
        add_0: 0 + ?a = ?a

  right_minus: ?a + - ?a = 0
   left_minus: - ?a + ?a = 0

   mult_assoc: ?a * ?b * ?c = ?a * (?b * ?c)

 distrib_left: ?a * (?b + ?c) = ?a * ?b + ?a * ?c
distrib_right: (?a + ?b) * ?c = ?a * ?c + ?b * ?c
THEOREM   

Zero is a right zero element of multiplication.

$$\forall x \in R : x0 = 0$$

Proof available

THEOREM   

Zero is a left zero element of multiplication.

$$\forall x \in R : 0x = 0$$

Proof available

THEOREM    \((-x) * y = -(x * y)\)

Proof available

THEOREM    \(x * (-y) = -(x * y)\)

Proof available

THEOREM    \((-x) * (-y) = x * y\)

Proof available

THEOREM    \(-(-x) = x\)

Proof available

Proper subsets

Proper supersets