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INTEGRAL DOMAINS AND FIELDS

Integral domain. A set S with two binary operations, called addition and multiplication, which have the following properties:

1. Closed under addition.

a,b S implies a + b S

2. Associative Law holds under addition.

a + (b + c) = (a + b) + c for any a,b,c in S

3. Additive Identity 0 exists under addition.

a + 0 = 0 + a = a for all a in S

4. Additive Inverse exists for every element under addition.

a + (-a) = (-a ) + a = 0 for all a in S

5. Commutative Law holds under addition.

a + b = b + a for all a,b in S

6. Closed under multiplication.

a,b S implies ab S

7. Associative Law holds under multiplication.

a(bc) = (ab)c for any a,b,c in S

8. Identity element 1 (i.e. unit element) exists under multiplication.

1a = a1 = a for all a in S

9. Commutative Law holds under multiplication.

ab = ba for all a,b in S

10. Left Distributive Law holds -- multiplication over addition

a(b+c) = ab + ac for all a,b,c in S

11. Right Distributive Law holds -- multiplication over addition

(a+b)c = ac + bc for all a,b,c in S

12. No proper divisors of zero (i.e. there are no two nonzero elements a and b for which ab = 0).

An integral domain is a commutative ring with unit element which has no proper divisors of zero.

Example. The set of all integers (positive, negative and 0) is an integral domain. The integers obey all the above axioms.

Subdomain. A subdomain of an integral domain D is a subset of D which is itself an integral domain with respect to the ring operations of D.

Ordered integral domain. An ordered integral domain is an integral domain D that contains a set of “positive” elements satisfying the conditions: 1) the sum and product of two positive elements is positive; 2) for a given element x of D, one and only one of the following is true: x is positive, x = 0 , -x is positive (where 0 is the additive identity and -x is the additive inverse of x).

Example. The set I of integers meet these requirements.

Characteristic of an integral domain. Because the elements of a ring form an additive group, each element of a ring generates under addition a cyclic group which is either finite of order n ≥1 or an infinite cyclic group. The order of this group is the order (or period) of the generating member. The characteristic of an integral domain is the additive order (or period) of the multiplicative identity 1.

Def. Unit. A unit in a integral domain is defined as an element having a multiplicative inverse. In other words it is an element u for which there is a v such that uv = 1. In the integral domain I of the integers the only integers that meet this requirement are the integers 1.

Def. Associate. An element b of an integral domain is called an associate of an element a if b = au where u is a unit. In the integral domain of the integers the only units are 1 and the associates of an integer a are the integers a.

Unique factorization domain. A unique factorization domain is an integral domain for which each member is a unit, or is a prime, or can be expressed as the product of a finite number of primes and this expression is unique except for unit factors and the order of the factors.

Example. The set I of integers is a unique factorization domain.

Theorems.

1] The Cancellation Law holds in an integral domain.

2] Let D be an integral domain and I be an ideal in D. Then D/I is an integral domain if and only if I is a prime ideal in D.

3] The characteristic of an integral domain is either zero or prime.

Field. A field is an integral domain which contains a multiplicative inverse for every nonzero element i.e. every element except the additive identity 0.

An equivalent definition can be given as follows. It is a set with two binary operations called addition and multiplication with the following properties:

(a) it is a commutative group with respect to addition.

(b) multiplication is commutative

(c) omitting the additive identity 0, it is a group with respect to multiplication

(d) a(b + c) = ab + ac for any a,b,c

Examples. The set of all rational numbers, the set of all real numbers, and the set of all complex numbers are all examples of fields.

To properly understand integral domains and fields one must realize that an integral domain is really just an axiomatic description of the basic properties of the integers and a field is really just an axiomatic description of the basic properties of the rational numbers. The basic difference between the integers and the rational numbers is that the rational numbers have multiplicative inverses and the integers do not.

1] An integral domain with a finite number of elements is a field.

2] The characteristic of a field is either zero or is a prime.

References.

James / James. Mathematics Dictionary.

Frank Ayres. Modern Algebra. (Schaum)

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