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The examples to keep in mind are these: the set of integers Z; the set Zn 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:
a+(b+c) = (a+b)+c and a·(b·c) = (a·b)·c.
a+b = b+a and a·b = b·a.
a·(b+c) = a·b + a·c and (a+b)·c = a·c + b·c.
a + 0 = a and 0 + a = a.
a + x = 0 and x + a = 0have 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. (Zn) The rings Zn 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
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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