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Mathematically, a group has
- closure: all resultant elements in group
- associativity: (a + b) + c = a + (b + c)
- identity: there is an element such that a + 0 = 0 + a = a
- inverse: for each element, a, there is an a' such that
a + a' = a' + a = 0 (with +, the inverse is the negative)
An Abelian group also has commutativity:
- commutative: for all a, b, a + b = b + a
A ring is a combination of two groups, one for each of two different kinds of operation, usually addition and multiplication.
So a ring is an Abelian group like that above,
plus the operation of multiplication
with the following laws for the second operation:
- closure: all resultant elements in group
- associativity: (a * b) * c = a * (b * c)
plus, a ring must have left and right distributive laws:
- left distributive: a * (b + c) = a * b + a * c
- right distributive: (a + b) * c = a * c + b * c
A ring with identity is a ring with an identity law for the second operation.
- identity: a * 1 = 1 * a = a
A commutative ring also has
- commutative: for all a, b, a * b = b * a
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