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