Cyclic Group

Cyclic Group

We define exponentiation within a group as repeated application of the group operator, so that a³ = a · a · a. Further, we define a⁰ = e, the identity element; and a^-n = (a')ⁿ. A group G is cyclic if every element of G is a power a^k (k is an integer) of a fixed element a G. The element a is said to generate the group G, or to be a generator of G. A cyclic group is always abelian, and may be finite or infinite.

The additive group of integers is an infinite cyclic group generated by the element 1. In this case, powers are interpreted additively, so that n is the nth power of 1.

Rings

A ring R, sometimes denoted by {R, +, x}, is a set of elements with two binary operations, called addition and multiplication,^[2] such that for all a, b, c in R the following axioms are obeyed:

^[2] Generally, we do not use the multiplication symbol, x, but denote multiplication by the concatenation of two elements.

(A1-A5) R is an abelian group with respect to addition; that is, R satisfies axioms A1 through A5. For the case of an additive group, we denote the identity element as 0 and the inverse of a as a.
(M1) Closure under multiplication:	If a and b belong to R, then ab is also in R.
(M2) Associativity of multiplication:	a(bc) = (ab)c for all a, b, c in R.
(M3) Distributive laws:	a(b + c) = ab + ac for all a, b, c in R. (a + b)c = ac + bc for all a, b, c in R.

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In essence, a ring is a set in which we can do addition, subtraction [a b = a + (-b)], and multiplication without leaving the set.

With respect to addition and multiplication, the set of all n-square matrices over the real numbers is a ring.

A ring is said to be commutative if it satisfies the following additional condition:

(M4) Commutativity of multiplication:	ab = ba for all a, b in R.

Let S be the set of even integers (positive, negative, and 0) under the usual operations of addition and multiplication. S is a commutative ring. The set of all n-square matrices defined in the preceding example is not a commutative ring.

Next, we define an integral domain, which is a commutative ring that obeys the following axioms:

(M5) Multiplicative identity:	There is an element 1 in R such that a1 = 1a = a for all a in R.
(M6) No zero divisors:	If a, b in R and ab = 0, then either a = 0 or b = 0.

Let S be the set of integers, positive, negative, and 0, under the usual operations of addition and multiplication. S is an integral domain.

Fields

A field F, sometimes denoted by {F, +, x}, is a set of elements with two binary operations, called addition and multiplication, such that for all a, b, c in F the following axioms are obeyed:

(A1M6) F is an integral domain; that is, F satisfies axioms A1 through A5 and M1 through M6.
(M7) Multiplicative inverse:	For each a in F, except 0, there is an element a-1 in F such that aa-1 = (a-1)a = 1.

In essence, a field is a set in which we can do addition, subtraction, multiplication, and division without leaving the set. Division is defined with the following rule: a/b = a(b^-1).

Familiar examples of fields are the rational numbers, the real numbers, and the complex numbers. Note that the set of all integers is not a field, because not every element of the set has a multiplicative inverse; in fact, only the elements 1 and -1 have multiplicative inverses in the integers.