Overview / review of commutative rings

For: Math 421 at Northern Illinois University
From: Beachy/Blair, Abstract Algebra, Second Edition
Covering: Sections 5.1 through 5.4

Commutative rings, in general

The examples you should have 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 we will use are the same as those for a field, with two crucial exceptions. We have dropped the requirement that each nonzero element has a multiplicative inverse, in order to include integers and polynomials in the class of objects we want to study.

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
(i) R is closed under addition and multiplication; and
(ii) if a is in R, then -a is in 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.

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

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 a^n = 0. Note that exercises in Section 1.4 contain information about idempotent and nilpotent elements in Zn. The group of units of Zn is also studied in Section 1.4.

Definition 5.2.1. Let R and S be commutative rings. A function f : R -> S is called a ring homomorphism if
f(a+b) = f(a) + f(b) and f(ab) = f(a) f(b) for all a,b in R.
A ring homomorphism that is one-to-one and onto is called an isomorphism. If there is an isomorphism from R onto S, we say that R is isomorphic to S. An isomorphism from the commutative ring R onto itself is called an automorphism of R.

Proposition 5.2.2. The inverse of a ring isomorphism is a ring isomorphism; the composition of two ring isomorphisms is a ring isomophism.

Proposition 5.2.3. Let f : R -> S be a ring homomorphism. Then
(a) f(0) = 0;
(b) f(-a) = -f(a) for all a in R;
(c) if R has an identity 1, then f(1) is idempotent;
(d) f(R) is a subring of S.

Definition 5.2.4. Let f : R -> S be a ring homomorphism. The set
{a in R | f(a) = 0 } is called the kernel of f, denoted by ker(f).

Proposition 5.2.5. Let f : R -> S be a ring homomorphism.
(a) If a,b are in ker(f) and r is in R, then a+b, a-b, and ra belong to ker(f).
(b) The homomorphism f is an isomorphism if and only if ker(f) = {0} and f(R) = S.

Example 5.2.5. Let R and S be commutative rings, let f : R -> S be a ring homomorphism, and let s be any element of S. Then there exists a unique ring homomorphism f# : R[x] -> S such that
f# (r) = f(r) for all r in R and f# (x) = s, defined by
f#(a0 + a1x + ... + amxm) = f(a0) + f(a1) s + ... + f(am) sm .

Proposition 5.2.7. Let R and S be commutative rings. The set of ordered pairs (r,s) such that r is in R and s is in S is a commutative ring under componentwise addition and multiplication.

Definition 5.2.8. Let R and S be commutative rings. The set of ordered pairs (r,s) such that r is in R and s is in S is called the direct sum of R and S.

Example 5.2.10. The ring Z /nZ is isomorphic to the direct sum of the rings Z /kZ that arise in the prime factorization of n. This describes the structure of Z /nZ in terms of simpler rings, and is the first example of what is usually called a ``structure theorem.'' This structure theorem can be used to determine the invertible, idempotent, and nilpotent elements of Z /nZ, and provides an easy proof of our earlier formula for the Euler phi-function in terms of the prime factors of n.

Definition 5.2.9. Let R be a commutative ring with identity. The smallest positive integer n such that (n)(1) = 0 is called the characteristic of R, denoted by char(R). If no such positive integer exists, then R is said to have characteristic zero.

Ideals and factor rings

Definition 5.3.1. Let R be a commutative ring. A nonempty subset I of R is called an ideal of R if
(i) I is a subgroup of R (under addition),
(ii) ra is in I, for all a in I and r in R.

Proposition 5.3.2. Let R be a commutative ring with identity. Then R is a field if and only if it has no proper nontrivial ideals.

Theorem 5.3.6. If I is an ideal of the commutative ring R, then R/I is a commutative ring, under the operations
(a+I)+(b+I) = (a+b)+I and (a+I)(b+I) = ab+I, for all a,b in R.

Proposition 5.3.7. Let I be an ideal of the commutative ring R.
(a) The natural projection mapping p : R -> R/I defined by p(a) = a+I for all a in R is a ring homomorphism, and ker(p) = I.
(b) There is a one-to-one correspondence between the ideals of R/I and ideals of R that contain I.

Theorem 5.2.6. [Fundamental Homomorphism Theorem for Rings] Let f : R -> S be a ring homomorphism. Then R/ker(f) is isomorphic to f(R).

Integral domains

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.

The ring of integers Z is the most fundamental example of an integral domain. The ring of all polynomials with real coefficients is also an integral domain, but the larger ring of all real valued functions is not an integral domain. The cancellation law for multiplication holds in R if and only if R has no nonzero divisors of zero. One way in which the cancellation law holds in R is if nonzero elements have inverses in a larger ring; the next two results characterize integral domains as subrings of fields (that contain 1).

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

Theorem 5.4.4. Let D be an integral domain. Then there exists a field F that contains a subring isomorphic to D.

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

Proposition 5.2.10. An integral domain has characteristic 0 or p, for some prime number p.

Definition 5.3.8. Let I be a proper ideal of the commutative ring R. Then I is said to be a prime ideal of R if for all a,b in R it is true that ab in I implies a in I or b in I.
The ideal I is said to be a maximal ideal of R if for all ideals J of R such that I is a subset of J and J is a subset of R, either J = I or J = R.

Proposition 5.3.9. Let I be a proper ideal of the commutative ring R with identity.
(a) The factor ring R/I is a field if and only if I is a maximal ideal of R.
(b) The factor ring R/I is a integral domain if and only if I is a prime ideal of R.
(c) If I is maximal, then it is a prime ideal.

Definition 5.3.3. Let R be a commutative ring with identity, and let a in R. The ideal
Ra = { x in R | x = ra for some r in R } is called the principal ideal generated by a.
An integral domain in which every ideal is a principal ideal is called a principal ideal domain.

Example 5.3.1. (Z is a principal ideal domain) Theorem 1.1.4 shows that the ring of integers Z is a principal ideal domain. Moreover, given any nonzero ideal I of Z, the smallest positive integer in I is a generator for the ideal.

Theorem 5.3.10. Every nonzero prime ideal of a principal ideal domain is maximal.

Example 5.3.7. (Ideals of F[x]) Let F be any field. Then F[x] is a principal ideal domain, since the ideals of F[x] have the form I = ( f(x) ), where f(x) is the unique monic polynomial of minimal degree in the ideal. The ideal I is prime (and hence maximal) if and only if f(x) is irreducible. If p(x) is irreducible, then the factor ring F[x]/( p(x) ) is a field.

Example 5.3.8. (Evaluation mapping) Let F be a subfield of E, and for any element u in E define the evaluation mapping fu : F[x] -> E by fu (g(x)) = g(u), for all g(x) in F[x]. Since fu (F[x]) is a subring of E that contains 1, it is an integral domain, and so the kernel of fu is a prime ideal. Thus if the kernel is nonzero, then it is a maximal ideal, so F[x]/ker(fu) is a field, and the image of fu is a subfield of E.

Rings vs. groups

      GROUPS                               RINGS

    Examples:                            Examples:
Symmetric group                      All n by n matrices
General linear group                 Polynomial rings

One binary operation                 Two binary operations +, .
    associative,                         abelian group under +
    identity element,                    mult is associative
    inverses                             distributive laws

Group homomorphisms                  Ring homomorphisms

    f(g)f(h) = f(gh)                     f(r)+f(s) = f(r+s)
                                          f(r)f(s) = f(rs)

Kernels of group homomorphisms       Kernels of ring homomorphisms 
    Normal subgroups:                    Ideals:
    gNg   contained in N              subgroups with rI and Ir contained in I

Factor groups                        Factor rings
    cosets gN, where N is normal         cosets r+I, where I is an ideal
    (gN)(hN) = gh N                      (r+I)+(s+I) = (r+s)+I
                                          (r+I)(s+I) = rs+I

Some classes of groups:              Corresponding classes of rings:

1. Abelian groups                    1. Integral domains 
    gh=hg, for all g,h                   rs=sr, for all r,s 
                                         rs = 0 implies r = 0 or s = 0

2. Cyclic groups                     2. Principal ideal domains 

    Z; Z/nZ                              Z; F[x] (F a field)

3. Simple abelian groups             3. Fields 

    Z/pZ                                Q; R; C; Z/pZ

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