12 Introduction to Rings 251 7. Show that the three properties listed in Exercise 6 are valid for Z where p is prime. 8. Show that a ring is commutative if it has the property that ab implies b c when a 0. 9. Prove that the intersection of any collection of subrings ofa ring R is a subring of R 10. Verify that Examples 8 through 13 in this chapter are as stated. 11. Prove rules 3 through 6 of Theorem 12.1. 12. Let a, b, and c be elements of a commutative ring, and suppose that is a unit. Prove that b divides c if and only if ab divides са 13. Describe all the subrings of the ring of integers. 14. Let a and b belong to a ring R and let m be an integer. Prove that m (ab) = (m a)b 15. Show that if m and n are integers and a and b are elements from a ring, then (m a)(n b) = (mn) (ab). (This exercise is referred to in Chapters 13 and 15.) 16. Show that if n is an integer and a is an element from a ring, then n (-a) = -(n a). 17. Show that a ring that is cyclic under addition is commutative. 18. Let a belong to a ring R. Let S {x E R | ax a subring of R. 19. Let R be a ring. The center of R is the set {x E Rlax = xa for all a in R}. Prove that the center of a ring is a subring. 20. Describe the elements of M, (Z) (see Example 4) that have multipli- cative inverses. = a(m b). = 0}. Show that S is R. are rings that contain nonzero ele- 21. Suppose that R,, R2, ments. Show that R, R, . .R has a unity if and only if each R, has a unity. 22. Let R be a commutative ring with unity and let U(R) denote the set of units of R. Prove that U(R) is a group under the multiplication of R. (This group is called the group of units of R.) 23. Determine U(Z[i]) (see Example 11). 24. If Ri, R2 U(R R, 25. Determine U(Z[x]). (This exercise is referred to in Chapter 17.) 26. Determine U(R[x]). n An R, are commutative rings with unity, show that R.) U(R) U(R) .UR). n 1' 27. Show that a unit of a ring divides every element of the ring. 28. In Z show that 4 1 2; in Za, show that 3 7; in Z, show that 9 | 12. LOUD

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12
Introduction to Rings
251
7. Show that the three properties listed in Exercise 6 are valid for Z
where p is prime.
8. Show that a ring is commutative if it has the property that ab
implies b c when a 0.
9. Prove that the intersection of any collection of subrings ofa ring R
is a subring of R
10. Verify that Examples 8 through 13 in this chapter are as stated.
11. Prove rules 3 through 6 of Theorem 12.1.
12. Let a, b, and c be elements of a commutative ring, and suppose that
is a unit. Prove that b divides c if and only if ab divides
са
13. Describe all the subrings of the ring of integers.
14. Let a and b belong to a ring R and let m be an integer. Prove that
m (ab) = (m a)b
15. Show that if m and n are integers and a and b are elements from a
ring, then (m a)(n b) = (mn) (ab). (This exercise is referred to
in Chapters 13 and 15.)
16. Show that if n is an integer and a is an element from a ring, then
n (-a) = -(n a).
17. Show that a ring that is cyclic under addition is commutative.
18. Let a belong to a ring R. Let S {x E R | ax
a subring of R.
19. Let R be a ring. The center of R is the set {x E Rlax = xa for all
a in R}. Prove that the center of a ring is a subring.
20. Describe the elements of M, (Z) (see Example 4) that have multipli-
cative inverses.
= a(m b).
= 0}. Show that S is
R. are rings that contain nonzero ele-
21. Suppose that R,, R2,
ments. Show that R, R, . .R has a unity if and only if
each R, has a unity.
22. Let R be a commutative ring with unity and let U(R) denote the set
of units of R. Prove that U(R) is a group under the multiplication of
R. (This group is called the group of units of R.)
23. Determine U(Z[i]) (see Example 11).
24. If Ri, R2
U(R R,
25. Determine U(Z[x]). (This exercise is referred to in Chapter 17.)
26. Determine U(R[x]).
n
An
R, are commutative rings with unity, show that
R.) U(R) U(R) .UR).
n
1'
27. Show that a unit of a ring divides every element of the ring.
28. In Z show that 4 1 2; in Za, show that 3 7; in Z, show that 9 | 12.
LOUD
Transcribed Image Text:12 Introduction to Rings 251 7. Show that the three properties listed in Exercise 6 are valid for Z where p is prime. 8. Show that a ring is commutative if it has the property that ab implies b c when a 0. 9. Prove that the intersection of any collection of subrings ofa ring R is a subring of R 10. Verify that Examples 8 through 13 in this chapter are as stated. 11. Prove rules 3 through 6 of Theorem 12.1. 12. Let a, b, and c be elements of a commutative ring, and suppose that is a unit. Prove that b divides c if and only if ab divides са 13. Describe all the subrings of the ring of integers. 14. Let a and b belong to a ring R and let m be an integer. Prove that m (ab) = (m a)b 15. Show that if m and n are integers and a and b are elements from a ring, then (m a)(n b) = (mn) (ab). (This exercise is referred to in Chapters 13 and 15.) 16. Show that if n is an integer and a is an element from a ring, then n (-a) = -(n a). 17. Show that a ring that is cyclic under addition is commutative. 18. Let a belong to a ring R. Let S {x E R | ax a subring of R. 19. Let R be a ring. The center of R is the set {x E Rlax = xa for all a in R}. Prove that the center of a ring is a subring. 20. Describe the elements of M, (Z) (see Example 4) that have multipli- cative inverses. = a(m b). = 0}. Show that S is R. are rings that contain nonzero ele- 21. Suppose that R,, R2, ments. Show that R, R, . .R has a unity if and only if each R, has a unity. 22. Let R be a commutative ring with unity and let U(R) denote the set of units of R. Prove that U(R) is a group under the multiplication of R. (This group is called the group of units of R.) 23. Determine U(Z[i]) (see Example 11). 24. If Ri, R2 U(R R, 25. Determine U(Z[x]). (This exercise is referred to in Chapter 17.) 26. Determine U(R[x]). n An R, are commutative rings with unity, show that R.) U(R) U(R) .UR). n 1' 27. Show that a unit of a ring divides every element of the ring. 28. In Z show that 4 1 2; in Za, show that 3 7; in Z, show that 9 | 12. LOUD
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