solve these following

17. Show that 0 is the only nilpotent element in an integral domain. 18. A ring element a is called an idempotent if a2 = a. Prove that the 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 - ab, a + b - ab, a+ b - 2ab. 20. Show that Z has a nonzero nilpotent element if and only if n is di- visible by the square of some prime. 21. Let R be the ring of real-valued continuous functions on [-1, 1]. Show that R has zero-divisors. 22. Prove that if a is a ring idempotent, then a" = a for all positive inte- gers n. 23. Determine all ring elements that are both nilpotent elements and idempotents. 24. Find a zero-divisor in Z[i] = (a + bila, b E Z;). 25. Find an idempotent in Z,[i] = (a + bila, b E Z;). 26. Find all units, zero-divisors, idempotents, and nilpotent elements in Z, OZ6. 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. 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 at least two elements. Prove that K is a subfield of F if, for any a, b (b + 0) in K, a - b and ab-1 belong to K. 30. Let d be a positive integer. Prove that ([Vd] = ta + bVd I 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 1. Show that P = {n . 1 In E 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