34 Rings 262 16. Show that the nilpotent elements of a commutative ring form а subring. 17. Show that 0 is the only nilpotent element in an integral domain 18. A ring element a is called an idempotent if a2 only idempotents in an integral domain are 0 and 1 19. Let a and b be idempotents in a commutative ring. Show that each of the following is also an idempotent: ab, a = a. Prove that the 35 3 ab, a + b - ab, 3 a+b-2ab. 20. Show that Z has a nonzero nilpotent element if and only if n is di. visible by the 3 n of some prime. square 21. Let R be the ring of real-valued continuous functions on [-1, 11 Show that R has zero-divisors. = a for all positive inte- 22. Prove that if a is a ring idempotent, then a" gers n. 23. Determine all ring elements that are both nilpotent elements and idempotents. 24. Find a zero-divisor in ZG[i] 25. Find an idempotent in Z,[i] 26. Find all units, zero-divisors, idempotents, and nilpotent elements in Z, Z 27. Determine all elements of a ring that are both units and idempotents. 28. Let R be the set of all real-valued functions defined for all real numbers under function addition and multiplication. = {a + bi | a, b E Z,}. = {a + bi | a, bE Z,}. 1 I1 3 a. Determine all zero-divisors of R. b. Determine all nilpotent elements of R. c. Show that every nonzero element is a zero-divisor or a unit. 29. (Subfield Test) Let F be a field and let K be a subset of F with least two elements . Prove that K is a subfield of F if, for any a, b (b 0) in K, a - b and ab belong to K. 30. Let d be a positive integer. Prove that Q[Vd] ={a + bVdl a, b E Q} is a field. 31. Let R be a ring with unity 1. If the product of any pair of nonzero elements of R is nonzero, prove that ab 1 implies ba = 1. 32. Let R = {0, 2, 4, 6, 8} under addition and multiplication modulo 10. Prove that R is a field. 33. Formulate the appropriate definition of a subdomain (that is, a "sub" integral domain). Let D be an integral domain with unity I. Show that P = {n 1 |nE Z) (that is, all integral multiples of 1) is a subdomain of D. Show that P is contained in every subdomain of D. What can we say about the order of P?