is a ring morphism, where f assigns to each linear transformation its standard matrix, that is, its n × n coefficient matrix with respect to the standard basis of Rn.

Solution. If α is a linear transformation from Rn to itself, then

Matrix addition and multiplication is defined so that f (α + β) = f (α) + f (β) and f (α Ž β) = f (α) · f (β). Also, if ι is the identity linear transformation then f (ι) is the identity matrix.

Any matrix defines a linear transformation, so that f is surjective. Furthermore, f is injective, because any matrix can arise from only one linear transformation. In fact, the j th column of the matrix must be the image of the

Proof. Since the function is defined in terms of representatives of equivalence classes, we first check that it is well defined. If [x]24 = [y]24,

then x ≡ y mod 24 and 24|(x − y). Hence 4|(x − y) and [x]4 = [y]4, which shows that f is well defined.

We now check the conditions for f to be a ring morphism.

(i)f ([x]24 + [y]24) = f ([x + y]24) = [x + y]4 = [x]4 + [y]4.

(ii)f ([x]24 · [y]24) = f ([xy]24) = [xy]4 = [x]4 · [y]4.

NEW RINGS FROM OLD

This section introduces various methods for constructing new rings from given rings. These include the direct product of rings, matrix rings, polynomial rings, rings of sequences, and rings of formal power series. Perhaps the most important class of rings constructible from given rings is the class of quotient rings. Their construction is analogous to that of quotient groups and is discussed in Chapter 10.

If (R, +, ·) and (S, +, ·) are two rings, their product is the ring (R × S, +, ·) whose underlying set is the cartesian product of R and S and whose operations are defined component-wise by

(r1, s1) + (r2, s2) = (r1 + r2, s1 + s2) and (r1, s1) · (r2, s2) = (r1 · r2, s1 · s2).

It is readily verified that these operations do indeed define a ring structure on R × S whose zero is (0R , 0S ), where 0R and 0S are the zeros of R and S, and whose multiplicative identity is (1R , 1S ), where 1R and 1S are the identities in R and S.

The product construction can be iterated any number of times. For example, (Rn, +, ·) is a commutative ring, where Rn is the n-fold cartesian product of R with itself.

Example 8.18. Write down the addition and multiplication tables for Z2 × Z3.

Solution. Let Z2 = {0, 1} and Z3 = {0, 1, 2}. Then Z2 × Z3 = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)}. The addition and multiplication tables are given in Table 8.6. In calculating these, it must be remembered that addition and multiplication are performed modulo 2 in the first coordinate and modulo 3 in the second coordinate.

We know that Z2 × Z3 and Z6 are isomorphic as groups; we now show that they are isomorphic as rings.

Theorem 8.19. Zm × Zn is isomorphic as a ring to Zmn if and only if gcd(m, n) = 1.

Proof. If gcd(m, n) = 1, it follows from Theorems 4.32 and 3.20 that the function

f : Zmn → Zm × Zn

TABLE 8.6. Ring Z2 × Z3

defined by f ([x]mn) = ([x]m, [x]n) is a group isomorphism. However, this function also preserves multiplication because

f([x]mn · [y]mn) = f ([xy]mn) = ([xy]m, [xy]n) = ([x]m[y]m, [x]n[y]n)

=([x]m, [x]n) · ([y]m, [y]n) = f ([x]mn) · f ([y]mn).

Also, f ([1]mn) = ([1]m, [1]n); thus f is a ring isomorphism.

It was shown in the discussion following Corollary 4.33 that if gcd(m, n) = 1, Zm × Zn and Zmn are not isomorphic as groups, and hence they cannot be isomorphic as rings.

We can extend this result by induction to show the following.

Theorem 8.20. Let m = m1 · m2 · · · mr , where gcd(mi , mj ) = 1 if i = j . Then Zm1 × Zm2 × · · · × Zmr is a ring isomorphic to Zm.

If R is a commutative ring, we can construct the ring of n × n matrices with entries from R, (Mn(R), +, ·). Addition and multiplication are performed as in real matrices.

For example, (Mn(Z2), +, ·) is the ring of n × n matrices with 0 and 1 entries. Addition and multiplication is performed modulo 2. This is a noncommutative ring with 2(n2 ) elements.

If R is a commutative ring, a polynomial p(x) in the indeterminate x over the ring R is an expression of the form

p(x) = a0 + a1x + a2x2 + · · · + anxn,

where a0, a1, a2, . . . , an R and n N. The element ai is called the coefficient of xi in p(x). If the coefficient of xi is zero, the term 0xi may be omitted, and if the coefficient of xi is one, 1xi may be written simply as xi .

Two polynomials f (x) and g(x) are called equal when they are identical, that is, when the coefficient of xn is the same in each polynomial for every n 0.

In particular,

a0 + a1x + a2x2 + · · · + anxn = 0

is the zero polynomial if and only if a0 = a1 = a2 = · · · = an = 0.

If n is the largest integer for which an = 0, we say that p(x) has degree n and write degp(x) = n. If all the coefficients of p(x) are zero, then p(x) is called the zero polynomial, and√ its degree is not defined.

For example, 4x2 − 3 is a polynomial over R of degree 2, ix4 − (2 + i)x3 + 3x is a polynomial over C of degree 4, and x7 + x5 + x4 + 1 is a polynomial over Z2 of degree 7. The number 5 is a polynomial over Z of degree 0; the zero

If the coefficient ring is not an integral domain, the degree of a product may be less than the sum of the degrees. For example, (2x3 + x) · (3x) = 3x2 in Z6[x].

Corollary 8.23. If R is an integral domain, so is R[x].

The construction of a polynomial ring can be iterated to obtain the ring of polynomials in n variables x1, . . . , xn, with coefficients from R. We define inductively R[x1, . . . , xn] = R[x1, . . . , xn−1][xn]. For example, consider a polynomial f in R[x, y] = R[x][y], say

f = f0 + f1y + f2y2 + · · · + fnyn,

where each fi = fi (x) is in R[x]. If we write fi = a0i + a1i x + a2i x2 · · · for each i, then

f (x, y) = a00 + a10x + a01y + a20x2 + a11xy + a02y2 + · · · .

Clearly, we can prove by induction from Corollary 8.22 that R[x1, . . . , xn] is an integral domain if R is an integral domain.

Proposition 8.24. Let R be a commutative ring and denote the infinite sequence of elements of R, a0, a1, a2, . . . , by ai . Define addition, +, and convolution,, of two such sequences by

ai + bi = ai + bi

and

Proof. Addition is clearly associative and commutative. The zero element is the zero sequence 0 = 0, 0, 0, . . . , and the negative of ai is −ai . Now

Similarly, bracketing the sequences in the other way, we obtain the same result, which shows that convolution is associative.

Convolution is clearly commutative and the distributive laws hold because

The identity in the ring of sequences is 1, 0, 0, . . . because

1, 0, 0, . . . a0, a1, a2, . . . = 1a0, 1a1 + 0a0, 1a2 + 0a1 + 0a0, . . .

= a0, a1, a2, . . . .

Therefore, (RN, +, ) is a commutative ring.

Suppose that aq and br are the first nonzero elements in the nonzero sequencesai and bi , respectively. Then the element in the (q + r)th position of their convolution is

Hence, if R is an integral domain, this element is not zero and the ring of sequences has no zero divisors.

The ring of sequences cannot be a field because 0, 1, 0, 0, . . . has no inverse.

In fact, for any sequence bi , 0, 1, 0, 0, . . . b0, b1, b2, . . . = 0, b0, b1, . . . , which can never be the identity in the ring.

A formal power series in x with coefficients from a commutative ring R is an expression of the form.

∞

a0 + a1x + a2x2 + · · · = ai xi where ai R.

i=0

In contrast to a polynomial, these power series can have an infinite number of nonzero terms.

We denote the set of all such formal power series by R[[x]]. The term formal is used to indicate that questions of convergence of these series are not considered. Indeed, over many rings, such as Zn, convergence would not be meaningful.

Motivated by RN, addition and multiplication are defined in R[[x]] by

∞ ∞ ∞

ai xi + bi xi = (ai + bi )xi

i=0 i=0 i=0