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.
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.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 30E: a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R....
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Topic Video
Question
42
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff82c0bab-f337-4781-985d-1029c419adb6%2F3eb7e28d-21cb-481a-b08a-7398f6b2e80e%2F19crk0n.jpeg&w=3840&q=75)
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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Elements Of Modern Algebra
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Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,