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.