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§ 5.1 Commutative Rings, Integral Domains

 
The examples to keep in mind are these: the set of integers Z; the set Z_n of integers modulo n; any field F (in particular the set Q of rational numbers and the set R of real numbers); the set F[x] of all polynomials with coefficients in a field F. The axioms are similar to those for a field, but the requirement that each nonzero element has a multiplicative inverse is dropped, in order to include integers and polynomials in the class of objects under study.

 
Definition 5.1.1 Let R be a set on which two binary operations are defined, called addition and multiplication, and denoted by + and ·. Then R is called a commutative ring with respect to these operations if the following properties hold:

(i) Closure: If a,b belong to R, then the sum a+b and the product a·b are uniquely defined and belong to R.

(ii) Associative laws: For all a,b,c in R,

a+(b+c) = (a+b)+c     and     a·(b·c) = (a·b)·c.

(iii) Commutative laws: For all a,b in R,

a+b = b+a     and     a·b = b·a.

(iv) Distributive laws: For all a,b,c in R,

a·(b+c) = a·b + a·c     and     (a+b)·c = a·c + b·c.

(v) Additive identity: The set R contains an additive identity element, denoted by 0, such that for all a in R,

a + 0 = a     and     0 + a = a.

(vi) Additive inverses: For each a in R, the equations

a + x = 0     and     x + a = 0

have a solution x in R, called the additive inverse of a, and denoted by -a.

a·1 = a     and     1·a = a.

 
As with groups, we will use juxtaposition to indicate multiplication, so that we will write ab instead of a·b.

 
Example 5.1.1. (Z_n) The rings Z_n form a class of commutative rings that is a good source of examples and counterexamples.

 
Definition 5.1.2 Let S be a commutative ring. A nonempty subset R of S is called a subring of S if it is a commutative ring under the addition and multiplication of S.

 
Proposition 5.1.3. Let S be a commutative ring, and let R be a nonempty subset of S. Then R is a subring of S if and only if

(i) R is closed under addition and multiplication; and

(ii) if a belongs to R, then -a belongs to R.

 
Definition 5.1.4. Let R be a commutative ring with identity element 1. An element a in R is said to be invertible if there exists an element b in R such that ab = 1.
    The element a is also called a unit of R, and its multiplicative inverse is usually denoted by a^-1.

 
An element e of a commutative ring R is said to be idempotent if e² = e. An element a is said to be nilpotent if there exists a positive integer n with aⁿ = 0.

 
Proposition 5.1.5. Let R be a commutative ring with identity. Then the set R^× of units of R is an abelian group under the multiplication of R.

 
Definition 5.1.6. A commutative ring R with identity is called an integral domain if for all a,b in R, ab = 0 implies a = 0 or b = 0.

 
Theorem 5.1.7 Let F be a field with identity 1. Any subring of F that contains 1 is an integral domain.

 
Theorem 5.1.8. Any finite integral domain must be a field.

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