Group

A group is a set of elements that has an associated operation that combines any two elements to create a third.  Like the set of integers and the addition operator.  However, a group is not that simple.  To be a valid group the following axioms must be true:

• Closure: The third element that is created by the operation must be an element of the set of elements contained in the group.
• Associativity: This means that (x + y) + z = k = x + (y + z) ie the grouping of the binary operations is irrelevant.
• Identity: The set of elements must contain an identity element (like 0 is for real numbers under addition).  By definition an identity is an element that won’t change the result when combined with other elements.  Like 0+4 = 4 or 1 * 4 = 4 or more general 0 + a = a or 0 * a = a.
• Invertability: This means that the the Latin square property holds.  This property says that given a group (G, *) and elements a and b elements of G, then there exists unique elements x and y such that a * x = b and a * y = b.

A special group the Abelian Group also has an additional axiom that it must satisfy, commutivity.  Commutivity is when the order the elements are combined is irrelevant, ie a * b = b * a must be true.

Modular Arithmetic

Modular arithmetic as the operation of a group is very important especially in cryptography.  Z/nZ is the group of integers modulo n (also Z/pZ is the group of integers mod p when p is a prime).

Ring

A Ring is a generalization of multiplication and addition.  By generalization this means being able to perform these arithmetic operations on things other than just plan numbers.  For example, functions, series, polynomials, etc.  More formally a Ring is a an abelian group with an additional binary operation that is distributive over the abelian group.  Also, the axioms of closure and associative must also hold.  However

Field

Is a commutitive Ring in which one can perform division (by any non-zero elements).