(b) Prove that S is a subring of M(R). [Hint: If B and Care in S, show that B + C and BC are in S by computing A(B + C) and A(BC).]
(b) Prove that S is a subring of M(R). [Hint: If B and Care in S, show that B + C and BC are in S by computing A(B + C) and A(BC).]
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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![Thomas W. Hungerford - Abstrac x
b My Questions | bartleby
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201..
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of R. [See Example 20.]
14. Let T be the ring in Example 8. Let S = {fe T|f(2) = 0}. Prove that S is a
subring of T.
15. Write out the addition and multiplication tables for
(a) Zz x Z3
(b) Z, x Z2
(e) Zz 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
elements of P(S) as follows:
9:15 AM
O Search for anything
EPIC
Ai
EPIC
O 4)
34
12/7/2020](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F8f876331-1794-40a5-a558-bed2903ebd57%2Fd05z95p_processed.png&w=3840&q=75)
Transcribed Image Text:Thomas W. Hungerford - Abstrac x
b My Questions | bartleby
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201..
...
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of 621
-- A Read aloud
V Draw
F Highlight
O Erase
77
IN U JUTIN5
of R. [See Example 20.]
14. Let T be the ring in Example 8. Let S = {fe T|f(2) = 0}. Prove that S is a
subring of T.
15. Write out the addition and multiplication tables for
(a) Zz x Z3
(b) Z, x Z2
(e) Zz 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
elements of P(S) as follows:
9:15 AM
O Search for anything
EPIC
Ai
EPIC
O 4)
34
12/7/2020
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