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8 RINGS AND FIELDS |
operations s and h can be used to explain Heaviside’s operational calculus, in which differential and integral operators are manipulated like algebraic symbols.
For further information on the algebraic aspects of generalized functions and distributions, see Erdelyi [36] or Marchand [37].
EXERCISES
8.1.Write out the tables for the ring Z4.
8.2.Write out the tables for the ring Z2 × Z2.
Which of the systems described in Exercises 8.3 to 8.12 are rings under addition and multiplication? Give reasons.
8.3. {a + b√ |
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√2 + c 3|a, b, c Z}.
8.6.{a + 3 2b|a, b Q}.
8.7.All 2 × 2 real matrices with zero determinant.
8.8.All rational numbers that can be written with denominator 2.
8.9.All rational numbers that can be written with an odd denominator.
8.10.(Z, +, ×), where + is the usual addition and a × b = 0 for all a, b Z.
8.11.The set A = {a, b, c} with tables given in Table 8.7.8.5. {a + b
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8.12. The set A = {a, b, c} with tables given in Table 8.8.
TABLE 8.8
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8.13. A ring R is called a boolean ring if a2 = a for all a R. (a) Show that (P (X), , ∩) is a boolean ring for any set X.
(b) Show that Z2 and Z2 × Z2 are boolean rings.
(c) Prove that if R is boolean, then 2a = 0 for all a R.
EXERCISES |
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Prove that any boolean ring is commutative. |
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ring where a b = (a b ) (a b).
(f) If (R, +, ·) is a boolean ring, show that (R, , , ) is a boolean algebra where a b = a · b, a b = a + b + a · b and a = 1 + a.
This shows that there is a one-to-one correspondence between boolean algebras and boolean rings.
8.14.If A and B are subrings of a ring R, prove that A ∩ B is also a subring of R.
8.15.Prove that the only subring of Zn is itself.
Which of the sets described in Exercises 8.16 to 8.20 are subrings of C? Give reasons.
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{0 + ib|b R}. |
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{a + b√ |
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{a + ib|a, b Z}. |
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8.17. {a + ib|a, b Q}.
8.19. {z C| |z| 1}.
Which of the rings described in Exercises 8.21 to 8.26 are integral domains and which are fields?
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Z2 × Z2. |
8.22. (P ({a}), , ∩). |
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{a + bi|a, b Q}. |
8.24. Z × R. |
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8.26. R[x]. |
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8.27.Prove that the set C(R) of continuous real-valued functions defined on the real line forms a ring (C(R), +, ·), where addition and multiplication of two functions f, g C(R) is given by
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Find all the zero divisors in the rings described in Exercises 8.28 to 8.33. |
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8.28. Z4. |
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8.29. Z10. |
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8.30. Z4 × Z2. |
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8.31. (P (X), , ∩). |
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8.32. M2(Z2). |
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8.33. Mn(R). |
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(R, +, ·) is commutative ring. |
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8.36. Show that H |
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is the conjugate |
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eralizes the complex numbers in the following way: If I = |
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178 |
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8 RINGS AND FIELDS |
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quaternion q has a unique representation in the form q |
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these relations determine the multiplication in H. If 0 = q H, show that |
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a noncommutative+ + |
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8.37.Find all the ring morphisms from Z to Z6.
8.38.Find all the ring morphisms from Z15 to Z3.
8.39.Find all the ring morphisms from Z × Z to Z × Z.
8.40.Find all the ring morphisms from Z7 to Z4.
8.41.If (A, +) is an abelian group, the set of endomorphisms of A, End(A), consists of all the group morphisms from A to itself. Show that (End(A), +, Ž )
is a ring under addition and composition, where (f + g)(a) = f (a) + g(a), for f, g End(A). This is called the endomorphism ring of A.
8.42.Describe the endomorphism ring End(Z2 × Z2). Is it commutative?
8.43.Prove that 10n ≡ 1 mod 9 for all n N. Then prove that an integer is divisible by 9 if and only if the sum of its digits is divisible by 9.
8.44.Find the number of nonisomorphic rings with three elements.
8.45. Prove that R[x] = R[y].
8.46. Prove that R[x, y] = R[y, x].
8.47. Let (R, +, ·) be a ring. Define the operations and Ž on R by r s = r + s + 1 and rŽs = r · s + r + s.
(a)Prove that (R, , Ž) is a ring.
(b)What are the additive and multiplicative identities of (R, , Ž)?
(c)Prove that (R, , Ž) is isomorphic to (R, +, ·).
8.48.Let a and b be elements of a commutative ring. For each positive integer n, prove the binomial theorem:
(a + b)n = an + |
1 an−1b + · · · + |
k an−k bk + · · · + bn. |
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8.49.Let (R, +, ·) be an algebraic object that satisfies all the axioms for a ring
except for the multiplicative identity. Define addition and multiplication in R × Z by
(a, n) + (b, m) = (a + b, n + m) and
(a, n) · (b, m) = (ab + ma + nb, nm).
Show that (R × Z, +, ·) is a ring that contains a subset in one-to-one correspondence with R that has all the properties of the algebraic object
(R, +, ·).
EXERCISES |
179 |
8.50.If R and S are commutative rings, prove that the ring of sequences (R × S)N is isomorphic to RN × SN.
8.51.If F is a field, show that the field of fractions of F is isomorphic to F .
8.52.Describe the field of fractions of the ring ({a + ib|a, b Z}, +, ·).
8.53.Let (S, ) be a commutative semigroup that satisfies the cancellation law; that is, a b = a c implies that b = c. Show that (S, ) can be embedded in a group.
8.54. Let T = {f : R → R|f (x) = a cos x + b sin x, a, b R}. Define addition of two such trigonometric functions in the usual way and define convolu-
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8.55. Let Tn = f : R → R|f (x) = |
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Show that (Tn, +, ) is a commutative ring where addition and convolution are defined as in Exercise 8.54. What is the multiplicative identity? Is the ring an integral domain?
8.56. If R is any ring, define R(i) = {a + bi|a, b R} to be the set of all formal
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ai = ia for all a R, then the ring axioms determine the addition and multiplication in R(i):
(r + si) + (r1 + s1i) = (r + r1) + (s + s1)i
(r + si)(r1 + s1i) = (rr1 − ss1) + (rs1 + sr1)i.
Thus, for example, R(i) = C.
(a)Show that R(i) is a ring, commutative if R is commutative.
(b)If R is commutative, show that a + bi is has an inverse in R(i) if and only if a2 + b2 has an inverse in R.
(c)Show that Z3(i) is a field of nine elements.
(d)Is C(i) a field? Is Z5(i) a field? Give reasons.
8.57.If R is a ring call e R an idempotent if e2 = e. Call R “tidy” if some positive power of every element is an idempotent.
(a)Show that every finite ring is tidy. [Hint: If a R, show that am+n = am for some n 1.]
(b)If R is tidy, show that uv = 1 in R implies that vu = 1.
(c)If R is a commutative tidy ring, show that every element of R is either invertible or a zero divisor.