14. Let Tbe the ring in Example 8. LetS=SET|f{2) = 0f. Prove that Sis a subring of T.
14. Let Tbe the ring in Example 8. LetS=SET|f{2) = 0f. Prove that Sis a subring of T.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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#14 on 3.1 on the picture I sent.
![po Thomas W. Hungerford - Abstrac X
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
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of 621
-- A Read aloud
V Draw
F Highlight
O Erase
77
(b) For any ring R, show that R* is a subring of R × R.
Caprite 2012Cmpe laing A Righe Rad Mng sot be peanor d whete or in part De toonie dee, d perty com y bapma tBodk d . Btalnta
dd ty eda d dany a te oa gapei C Lang righto ddenal a g teit gt m i
3.1 Definition and Examples of Rings
55
10. Is S= {(a, b)| a + b = 0} a subring of Z x Z? Justify your answer.
11. Let S be the subset of M(R) consisting of all matrices of the form
(a) Prove that S is a ring.
G ) is a right identity in S (meaning that AJ = A for
(b) Show that J =
every A in S).
(c) Show that Jis not a left identity in S by finding a matrix B in S such that
JB + B.
For more information about S, see Exercise 41.
12. Let Z[J denote the set {a + bi a, beZ}. Show that Z[] is a subring of C.
13. Let Z[V2] denote the set {a + b21a, beZ}. Show that Z[V2] is a subring
of R. See Example 20.]
14. Let Tbe the ring in Example 8. Let S = {feT|f(2) = 0}. Prove that S is a
subring of T.
15. Write out the addition and multiplication tables for
(a) Z, x Z;
(b) Z, × Z,
(c) Z, x Z,
16. Let A =
and 0 =
in M(R). Let S be the set of all matrices B
such that AB = 0.
(a) List three matrices in S. [Many correct answers are possible.]
(b) Prove that S is a subring of M(R). [Hint: If B and Care in S, show that
B + Cand BC are in S by computing A(B + C) and A(BC).]
17. Define a new multiplication in Z by the rule: ab = 0 for all a, b,eZ Show that
with ordinary addition and this new multiplication, Z is a commutative ring.
18. Define a new multiplication in Z by the rule: ab = 1 for all a, b,EZ. With
ordinary addition and this new multiplication, is Z is a ring?
19. Let S = {a, b, c} and let P(S) be the set of all subsets of S; denote the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F0770a08c-d101-4cad-972f-511aeee61219%2Fnbtk6f_processed.png&w=3840&q=75)
Transcribed Image Text:po Thomas W. Hungerford - Abstrac X
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
...
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of 621
-- A Read aloud
V Draw
F Highlight
O Erase
77
(b) For any ring R, show that R* is a subring of R × R.
Caprite 2012Cmpe laing A Righe Rad Mng sot be peanor d whete or in part De toonie dee, d perty com y bapma tBodk d . Btalnta
dd ty eda d dany a te oa gapei C Lang righto ddenal a g teit gt m i
3.1 Definition and Examples of Rings
55
10. Is S= {(a, b)| a + b = 0} a subring of Z x Z? Justify your answer.
11. Let S be the subset of M(R) consisting of all matrices of the form
(a) Prove that S is a ring.
G ) is a right identity in S (meaning that AJ = A for
(b) Show that J =
every A in S).
(c) Show that Jis not a left identity in S by finding a matrix B in S such that
JB + B.
For more information about S, see Exercise 41.
12. Let Z[J denote the set {a + bi a, beZ}. Show that Z[] is a subring of C.
13. Let Z[V2] denote the set {a + b21a, beZ}. Show that Z[V2] is a subring
of R. See Example 20.]
14. Let Tbe the ring in Example 8. Let S = {feT|f(2) = 0}. Prove that S is a
subring of T.
15. Write out the addition and multiplication tables for
(a) Z, x Z;
(b) Z, × Z,
(c) Z, x Z,
16. Let A =
and 0 =
in M(R). Let S be the set of all matrices B
such that AB = 0.
(a) List three matrices in S. [Many correct answers are possible.]
(b) Prove that S is a subring of M(R). [Hint: If B and Care in S, show that
B + Cand BC are in S by computing A(B + C) and A(BC).]
17. Define a new multiplication in Z by the rule: ab = 0 for all a, b,eZ Show that
with ordinary addition and this new multiplication, Z is a commutative ring.
18. Define a new multiplication in Z by the rule: ab = 1 for all a, b,EZ. With
ordinary addition and this new multiplication, is Z is a ring?
19. Let S = {a, b, c} and let P(S) be the set of all subsets of S; denote the
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