Excerpted from Abstract Algebra II, copyright 1996 by John Beachy
Sections 5.5 through 5.8
Localization in integral domains
Noncommutative examples
Isomorphism theorems

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It is important to note that we will always assume that our rings have a multiplicative identity element. Furthermore, in what represents a significant change from the previous sections, we will assume that a subring always has the same multiplicative identity as the ring itself, and that ring homomorphisms preserve the multiplicative identity element.

Localization in integral domains

5.8.9. Proposition. Let D be an integral domain with quotient field Q(D), and let P be a prime ideal of D. Then
DP = { ab-1 Q(D) | b P }
is an integral domain with D DP Q(D).

5.8.10. Definition. Let D be an integral domain with quotient field Q(D), and let P be a prime ideal of D. Then
DP = { ab-1 Q(D) | b P }
is called the localization of D at P.
If I is any ideal of D, then
IP = { ab-1 Q(D) | a I and b P }
is called the extension of I to DP.

5.8.11. Theorem. Let D be an integral domain with quotient field Q(D), and let P be a prime ideal of D.
(a) If J is any proper ideal of DP, then there exists an ideal I of D with I P such that J = IP. If J is a prime ideal, then so is I.
(b) If J is any prime ideal of D with J P, then JP is a prime ideal of DP with D JP = J.
(c) There is a one-to-one correspondence between prime ideals of DP and prime ideals of D that are contained in P. Furthermore, PP is the unique maximal ideal of DP.

Noncommutative examples

We want to include, among other examples, the study of n×n matrices. Recall that if F is a field, then the set of n×n matrices Mn(F) corresponds to the set of linear transformations of an n-dimensional vector space over F. This is a special case of the most general example of a ring with identity. Just as permutation groups are the generic groups (as shown by Cayley's theorem), the generic examples of rings are found in studying endomorphisms of abelian groups.

Example. 5.5.1. (Endomorphisms of abelian groups)
Let A be an abelian group, with its operation denoted by +. Let R be the set of all endomorphisms of A. That is, R is the set of all group homomorphisms f:A->A. We can define addition and multiplication of elements of R as follows: if f,g R, then
(f+g) (x) = f(x) + g(x) and (f . g) (x) = f(g(x))
for all x A.

We need to repeat Definition 5.1.1, dropping the assumption that multiplication is commutative. We will, however, assume that all rings have a multiplicative identity element. It can then be shown that the set of all endomorphisms of an abelian group A, denoted by End(A), forms a ring.

Definition A ring R is a set with two binary operations, denoted by addition and multiplication, such that the following properties hold:
(i) For all a,b,c R, a+(b+c) = (a+b)+c and a(bc) = (ab)c.
(ii) For all a,b R, a+b = b+a.
(iii) For all a,b,c R, a(b+c) = ab+ac and (a+b)c = ac+bc.
(iv) The set R contains an additive identity element, denoted by 0, and a multiplictive identity element, denoted by 1, such that a+0 = a, 1a = a, and a1=a, for all a R.
(v) For each a R, the equation a+x = 0 has a solution x = -a in R, the additive inverse of a.

Example. 5.5.2. (Polynomial Rings)
Let R be any ring. We let R[x] denote the set of infinite tuples
such that ai R for all i, and ai 0 for only finitely many terms ai. Two sequences are equal if and only if all corresponding terms are equal. We introduce addition and multiplication as follows:
(a0,a1,a2,...) + (b0,b1,b2,...) = (a0+b0,a1+b1,a2+b2,...)
(a0,a1,a2,...) . (b0,b1,b2,...) = (c0,c1,c2,...),
where cn = ai bn-i.
With these operations it can be shown that R[x] is a ring.
We can identify a R with (a,0,0,...) R[x], and so if R has an identity 1, then (1,0,0,...) is an identity for R[x]. If we let x=(0,1,0,...), then the elements of R[x] can be expressed in the form
a0 + a1 x + . . . + am-1 xm-1 + am xm,
allowing us to use our previous notation for the ring of polynomials over R in the indeterminate x. Note that although the elements of R need not commute with each other, they do commute with the indeterminate x.
If n is the largest nonnegative integer such that an 0, then we say that the polynomial has degree n, and an is called the leading coefficient of the polynomial.

Example. 5.5.3. (Differential operator rings)
Consider the homogeneous linear differential equation
an(x) Dn y + . . . + a1(x) D y + a0(x) y = 0,
where the solution y(x) is a polynomial with complex coefficients, and the terms ai(x) also belong to C[x]. The equation can be written in compact form as L(y)=0, where L is the differential operator
an(x) Dn + . . . + a1(x) D + a0(x),
with D = d/dx. Thus the differential operator can be thought of as a polynomial in the two indeterminates x and D, but in this case the indeterminates do not commute, since
D(x y(x)) = y(x) + x D(y(x)),
yielding the identity
Repeated use of this identity makes it possible to write the composition of two differential operators in the standard form
a0(x) + a1(x) D + . . . + an(x) Dn,
and we denote the resulting ring by C[x][D].

Example. 5.5.4. (Group algebras)
Let K be a field, and let G be a finite group of order n, with elements 1=g1, g2, . . . , gn. The group algebra KG is defined to be the n-dimensional vector space over K with the elements of G as a basis. Vector addition is used as the addition in the ring. Elements of KG can be described as sums of the form
ci gi
and multiplication is defined as for polynomials, where the product gi gj is given by the product in G.

Example. 5.5.5. (Matrix rings)
Let R be a ring. We let Mn(R) denote the set of all n×n matrices with entries in R.
For [aij] and [bij] in Mn(R), we use componentwise addition
[aij] + [bij] = [aij+bij]
and the multiplication is given by
[ajk] [bjk] = [cjk],
where [cij] is the matrix whose j,k-entry is
cjk = aji bik.

5.5.3. Definition. Let R be a ring with identity 1, and let a R. If ab=0 for some nonzero b R, then a is called a left zero divisor. Similarly, if ba=0 for some nonzero b R, then a is called a right zero divisor. If a is neither a left zero divisor nor a right zero divisor, then a is called a regular element.
The element a R is said to be invertible if there exists an element b R such that ab=1 and ba=1. The element a is also called a unit of R, and its multiplicative inverse is usually denoted by a-1.
The set of all units of R is denoted by R×.

5.5.4. Proposition. Let R be a ring. Then the set R× of units of R is a group under the multiplication of R.

5.5.5. Definition. A ring R in which each nonzero element is a unit is called a division ring or skew field.

Example. 5.5.8. (The quaternions)
The following subset of M2(C) is called the set of quaternions, and provides the best known example of a division ring that is not a field.
Q = a + b + c + d
See Example 3.3.7 for the group of quaternion units.

Isomorphism theorems

5.6.1. Definition. Let R be a ring. A nonempty subset I of R is called a left ideal of R if
(i) a b I for all a,b I and (ii) ra I for all a I and r R.
The subset I is called a right ideal of R if
(i) a b I for all a,b I and (ii) ar I for all a I and r R.
The subset I is called a two-sided ideal or simply an ideal of R if it is both a left ideal and a right ideal.

For any ring R, it is clear that the set {0} is an ideal, which we will refer to as the trivial ideal. Another ideal of R is the ring R itself.

5.6.2. Definition. Let R be a ring, and let a R. The left ideal
Ra = { x R | x = ra for some r R }
is called the principal left ideal generated by a.

5.6.3. Proposition. Let R be a ring, and let I,J be left ideals of R. The following subsets of R are left ideals.
(a) I J;
(b) I + J = { x R | x = a + b for some a I, b J };
(c) IJ = { ai bi | ai I, bi J, n Z+ }.

Example. 5.6.1. (Ideals of Mn(R))
Let R be a ring, and let Mn(R) be the ring of matrices over R. If I is an ideal of R, then the set Mn(I) of all matrices with entries in I is an ideal of S. Conversely, every ideal of S is of this type.

5.6.4. Theorem. If I is an ideal of the ring R, then R/I is a ring.

5.6.5. Definition. Let I be an ideal of the ring R. With the following addition and multiplication for all a,b R, the set of cosets
{ a+I | a R } is denoted by R/I, and is called the factor ring of R modulo I.
(a+I) + (b+I) = (a+b) + I and (a+I)(b+I) = ab + I

Let I be a proper ideal of the ring R. Then I is said to be a completely prime ideal of R if for all a,b R it is true that ab I implies a I or b I.

As in the commutative case, a ring is called a domain if (0) is a completely prime ideal. An element c of R is said to be regular if xc = 0 or cx = 0 implies x = 0, for all x R. Thus a ring is a domain if and only if every nonzero element is regular.

5.7.1. Definition. Let R and S be rings. A function :R->S is called a ring homomorphism if
(i) (a+b) = (a) + (b), for all a,b R,
(ii) (ab) = (a) (b), for all a,b R, and
(iii) (1) = 1.
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, and write R S. An isomorphism from the ring R onto itself is called an automorphism of R.

5.7.2. Proposition. Any ring R is isomorphic to a subring of an endomorphism ring End(A), for some abelian group A.

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

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

5.7.6. Proposition. Let R and S be rings, let :R->S be a ring homomorphism, and let
:{ x1, x2, . . . , xn}->Z(S) be any mapping into the center of S. Then there exists a unique ring homomorphism
: R [ x1, x2, . . . , xn] -> S such that
(r) = (r) for all r R and
(xi) = (xi), for i=1,2,...,n.

Example. 5.7.2. Let G and H be finite groups, and let K be a field. If :G->H is a group homomorphism, we can extend the mapping to a ring homomorphism :KG->KH as follows:
( x G cx x ) = x G cx (x).

5.7.7. Theorem. [Fundamental Homomorphism Theorem for Rings] Let :R->S be a ring homomorphism. Then (R) is a subring of S, R/ker() is a ring, and (R)R/ker().

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

5.7.9. Theorem. [Chinese Remainder Theorem] Let R be a ring, and let I1, I2 be ideals of R such I1+I2=R. Then
( R / I1) ( R / I2) R / (I1 I2).

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