From: Beachy/Blair,

Covering: Sections 5.1 through 5.4

**
Example 5.1.1. (Z_{n})
**
The rings

**
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 *e*^{2} = *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
**Z**_{n}.
The group of units of
**Z**_{n}
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#(a_{0} + a_{1}x + ... + a_{m}x^{m})
= f(a_{0}) + f(a_{1}) s + ... + f(a_{m}) s^{m} .

**
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* /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**.

(i)

(ii)

**
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)*.

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

**
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

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

**
Example 5.3.7. (Ideals of F[x])
**
Let

**
Example 5.3.8. (Evaluation mapping)
**
Let *F* be a subfield of *E*,
and for any element *u* in *E* define the evaluation mapping
*f _{u} : F*[x] ->

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: -1 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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