(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).]

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Chapter2: Second-order Linear Odes
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#16 part b

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
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.. ... Flash Player will no longer be supported after December 2020. Turn off Learn more 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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