# 42 Transcribed Image Text: 308 Rings 252 29. Suppose that a and b belong to a commutative ring R with unity. If 30. Suppose that

42 Transcribed Image Text: 308

Rings

252

29. Suppose that a and b belong to a commutative ring R with unity. If

30. Suppose that there is an integer n> 1 such that x”

x of some ring. If m is a positive integer

x for all elements

a is a unit of R and b2 0, show that a + b is a unit of R.

O for some a, show

and a”

0.

that a

31. Give an example of ring elements a and b with the properties that

ab 0 but ba 0.

х,

32. Let n be an integer greater than 1. In a ring in which x” = x for all

0.

0 implies ba

show that ab

33. Suppose that R is a ring such that x3 = x for allx in R. Prove that

0 for all x in R

бх

= a. Prove that a2n

a2

34. Suppose that a belongs to a ring and a

for all n 1.

= a for all a in Z. Do the same

6

35. Find an integer n > 1 such that a”

for Zo Show that no such n exists for Z when m is divisible by the

10

т

of some prime.

square

36. Let m and n be positive integers and letk be the least common mul-

tiple of m and n. Show that mZn nZ kZ.

37. Explain why every subgroup of Z, under addition is also a subring

of Zg

38. Is Z a subring of Z,?

39. Suppose that R is a ring with unity 1 and a is an element of R such

that a2 = 1. Let S

Co

ara l rE R}. Prove that S is a subring of R.

Does S contain 1?

40. Let M,(Z) be the ring of all 2 X 2 matrices over the integers and let R =

{, ;]z}

4.bEZ

a + b

a

Prove or disprove that R is a subring

a +b b

of M2(Z).

41. Let M2(Z) be the ring of all 2 X 2 matrices over the integers and let R

S

а

a

a, bE Z. Prove or disprove that R is a subring

a -b

b

of M(Z).

42. Let R

a

Prove or disprove that R is a subring

b

of M,(Z).

43. Let R Zze Z and S {(a, b, c) E Rla+ b = c}. Prove or

disprove that S is a subring of R.

44. Suppose that there is a positive even integer n such that a” = a for

all elements a of some ring. Show that -a a for all a in the ring.

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