Groups, Semi-Rings, Rings and Fields
February 10, 2017
Group
A group G is a set with one binary operations · : G × G → G such that,
(M1) (Associativity of multiplication) For any k, m, n in G,
k · (m · n) = (k · m) · n.
(M3) (Multiplicative Identity) For any n,
1 · n = n · 1 = n.
(M4) (Multiplicative Inverses) For any n ∈ G, with n 6= 0, there exists n−1 ∈ G such that
n−1 · n = n · n−1 = 1
We say that G is abelian if · also satisfies
(M2) (Commutativity of multiplication) For any n, m in G,
m · n = n · m.
Semi-Ring
A (commutative) semi-ring S is a set with two binary operations ·, + : S × S → S such that,
(A1) (Associativity of addition) For any k, m, n in S,
k + (m + n) = (k + m) + n.
1
(A2) (Commutativity for addition) For any n, m in S,
m + n = n + m.
(A3) (Additive Identity) There exists 0 ∈ S such that for any n ∈ S.
0 + n = n + 0 = n.
(M0) (0 Annihilates) For any n in S,
0 · n = n · 0 = 0.
(M1) (Associativity of multiplication) For any k, m, n in S,
k · (m · n) = (k · m) · n.
(M2) (Commutativity of multiplication) For any n, m in S,
m · n = n · m.
(M3) (Multiplicative Identity) For any n,
1 · n = n · 1 = n.
(D) (Distributivity) For any k, m, n in S,
k · (m + n) = (k · m) + k · n.
Ring
A (commutative) ring R is a set with two binary operations ·, + : R × R → R such that,
(A1) (Associativity of addition) For any k, m, n in R,
k + (m + n) = (k + m) + n.
2
(A2) (Commutativity for addition) For any n, m in R,
m + n = n + m.
(A3) (Additive Identity) There exists 0 ∈ R such that for any n ∈ R.
0 + n = n + 0 = n.
(A4) (Additive Inverses) For any n ∈ R, there exists −n ∈ R
(−n) + n = n + (−n) = 0.
(M1) (Associativity of multiplication) For any k, m, n in R,
k · (m · n) = (k · m) · n.
(M2) (Commutativity of multiplication) For any n, m in R,
m · n = n · m.
(M3) (Multiplicative Identity) For any n,
1 · n = n · 1 = n.
(D) (Distributivity) For any k, m, n in R,
k · (m + n) = (k · m) + k · n.
Remark. We did not assume (M0) for rings! Once we add (A4), it follows from the other axioms.
Field
A field F is a set with two binary operations ·, + : F × F → F such that,
3
(A1) (Associativity of addition) For any k, m, n in F ,
k + (m + n) = (k + m) + n.
(A2) (Commutativity for addition) For any n, m in F ,
m + n = n + m.
(A3) (Additive Identity) There exists 0 ∈ F such that for any n ∈ F .
0 + n = n + 0 = n.
(A4) (Additive Inverses) For any n ∈ F , there exists −n ∈ F
(−n) + n = n + (−n) = 0.
(M1) (Associativity of multiplication) For any k, m, n in F ,
k · (m · n) = (k · m) · n.
(M2) (Commutativity of multiplication) For any n, m in F ,
m · n = n · m.
(M3) (Multiplicative Identity) For any n,
1 · n = n · 1 = n.
(M4) (Multiplicative Inverses) For any n ∈ F , with n 6= 0, there exists n−1 ∈ F such that
n−1 · n = n · n−1 = 1
4
(D) (Distributivity) For any k, m, n in F ,
k · (m + n) = (k · m) + k · n.
(F) The additive identity 0 and the multiplicative identity 1 are not equal.
Ordered Field
A ordered ring is a ring R with a strict total order relation ≺ which satisfies
(STO1) (Trichotomy) For all x, y ∈ X exactly one of the following holds: x ≺ y, y ≺ x or x = y.
(STO2) (Transitivity) For all x, y, z ∈ X, if x ≺ y and y ≺ z then x ≺ z.
(STO3) (Monotony of addition) If m < n, then m + k < n + k.
(STO4) (Monotony of multiplication) If 0 < k, then n · k < m · k.
An ordered field is a field F which is ordered as a ring.
5