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Axioms satisfied by rings, integral domains and fields

A list of 15 axioms follows. Subsets of these 15 axioms are satisfied by rings, integral domains and fields.

There are three columns to the left of the axioms containing the abbreviations F, ID, and R which stand for Field, Integral Domain and Ring, respectively. If an axiom is satisfied by, say, a field then an F will appear to the left of the axiom. Likewise for integral domains and rings. Thus an F to the left of Axiom 1 means that Axiom 1 holds for a field.

For every a,b,c in a set S with two operations called

addition and multiplication:

Addition.

F ID R 1. Closed under addition.

F ID R a,b in S implies a + b in S

F ID R

F ID R 2. Associative Law holds under addition.

F ID R a + (b + c) = (a + b) + c

F ID R

F ID R 3. Identity element called 0 exists under addition.

F ID R a + 0 = 0 + a = a for all a in S

F ID R

F ID R 4. Inverse exists for every element under addition.

F ID R a + (-a) = (-a ) + a = 0 for all a in S

F ID R

F ID R 5. Commutative Law holds under addition.

F ID R a + b = b + a

F ID R

F ID R 6. Cancellation Law holds under addition.

F ID R a + x = b + x implies a = b

Multiplication.

F ID R 7. Closed under multiplication.

F ID R a,b in S implies ab in S

F ID R

F ID R 8. Associative Law holds under multiplication.

F ID R a(bc) = (ab)c

F ID R

F ID 9. Identity element called 1 exists under multiplication.

F ID 1a = a1 = a for all a in S

F ID

F 10. Inverse exists for every nonzero element under multiplication.

F aa^{-1} = a^{-1}a = 1 for all a in S (element 0 is excluded)

F

F ID 11. Commutative Law holds under multiplication.

F ID ab = ba

F ID

F ID R 12. Left Distributive Law holds -- multiplication over addition

F ID R a(b+c) = ab + ac

F ID R

F ID R 13. Right Distributive Law holds -- multiplication over addition

F ID R (a+b)c = ac + bc

F ID R

F ID 14. No proper divisors of zero (i.e. there are no nonzero members a and b for

F ID which ab = 0).

F ID

F ID 15. Cancellation Law under multiplication holds.

F ID ax = bx implies a = b

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