RINGS

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

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

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 D_P, then there exists an ideal I of D with I P such that J = I_P. If J is a prime ideal, then so is I.
(b) If J is any prime ideal of D with J P, then J_P is a prime ideal of D_P with D J_P = J.
(c) There is a one-to-one correspondence between prime ideals of D_P and prime ideals of D that are contained in P. Furthermore, P_P is the unique maximal ideal of D_P.

Noncommutative examples

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
(a₀,a₁,a₂,...)
such that a_i R for all i, and a_i 0 for only finitely many terms a_i. 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 c_n = a_i b_n-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 + . . . + a_m-1 x^m-1 + a_m x^m,
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 a_n 0, then we say that the polynomial has degree n, and a_n is called the leading coefficient of the polynomial.

Example. 5.5.3. (Differential operator rings)
Consider the homogeneous linear differential equation
a_n(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 a_i(x) also belong to C[x]. The equation can be written in compact form as L(y)=0, where L is the differential operator
a_n(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
Dx=1+xD.
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 + . . . + a_n(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, . . . , g_n. 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
c_i g_i
and multiplication is defined as for polynomials, where the product g_i g_j is given by the product in G.

Example. 5.5.5. (Matrix rings)
Let R be a ring. We let M_n(R) denote the set of all n×n matrices with entries in R.
For [a_ij] and [b_ij] in M_n(R), we use componentwise addition
[a_ij] + [b_ij] = [a_ij+b_ij]
and the multiplication is given by
[a_jk] [b_jk] = [c_jk],
where [c_ij] is the matrix whose j,k-entry is
c_jk = a_ji b_ik.

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

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 = { a_i b_i | a_i I, b_i J, n Z⁺ }.

Example. 5.6.1. (Ideals of M_n(R))
Let R be a ring, and let M_n(R) be the ring of matrices over R. If I is an ideal of R, then the set M_n(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, . . . , x_n}->Z(S) be any mapping into the center of S. Then there exists a unique ring homomorphism
: R [ x₁, x₂, . . . , x_n] -> S such that
(r) = (r) for all r R and
(x_i) = (x_i), 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} c_x x ) = _{x G} c_x (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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