Bomorpain 7. Prove that Risisomorphic to the ring Sof all 2 x 2 matrices of the form where a ER.
Bomorpain 7. Prove that Risisomorphic to the ring Sof all 2 x 2 matrices of the form where a ER.
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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3.3 #7 on the picture I sent.
![Thomas W. Hungerford - Abstrac x
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of 621
-- A) Read aloud
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F Highlight
O Erase
102
2. Use tables to show that Z, x Z, is isomorphic to the ring Rof Exercise 2 in
Section 3.1.
3. Let R be a ring and let.
of the form (a, a). Show that the function f.R→ R* given by f(a) = (a, a) is an
isomorphism.
be the subring of RX R consisting of all elements
4. Let Sbe the subring {0, 2, 4, 6, 8} of Z1, and let Z, = {0, 1,2, 3, 4,} (notation
as in Example 1). Show that the following bijection from Z, to Sis not an
isomorphism:
0→0 I-→2 2→4 3 →6 4→ 8.
5. Prove that the field R of real numbers is isomorphic to the ring of all 2 x 2
matrices of the form
,with a ER. [Hint: Consider the function f given
Courem
by f(a) =
6. Let Rand Sbe rings and let R be the subring of R × Sconsisting of all
elements of the form (a, Os). Show that the function f.R→R given by
f(a) = (a, Os) is an isomorphism.
7. Prove that Ris isomorphic to the ring S of all 2 × 2 matrices of the form
, where a ER.
8. Let Q(V2) be as in Exercise 39 of Section 3.1. Prove that the function
f:Q(V2) → Q(V2) given by f(a + bV2) = a – bV2 is an isomorphism.
9. If f:Z→Z is an isomorphism, prove that f is the identity map. [Hint: What
are f(1), f(1 + 1), ...?]
10. If R is a ring with identity and f:
ring S, prove that f(1r) is an idempotent in S. [Idempotents were defined in
Exercise 3 of Section 3.2.]
S is a homomorphism from R to a
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dd tg d d ny hate oa ingp Cn lning right dol c t da
3.3 Isomorphisms and Homomorphisms
81
11. State at least one reason why the given function is not a homomorphism.
(a) f:R→R and f(x) = Vx.
(b) g:E→E, where E is the ring of even integers and f(x) = 3x.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F0eb16f3b-1beb-4bf7-918b-c5390c3cebdb%2Fp38e9jg_processed.png&w=3840&q=75)
Transcribed Image Text:Thomas W. Hungerford - Abstrac x
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of 621
-- A) Read aloud
V Draw
F Highlight
O Erase
102
2. Use tables to show that Z, x Z, is isomorphic to the ring Rof Exercise 2 in
Section 3.1.
3. Let R be a ring and let.
of the form (a, a). Show that the function f.R→ R* given by f(a) = (a, a) is an
isomorphism.
be the subring of RX R consisting of all elements
4. Let Sbe the subring {0, 2, 4, 6, 8} of Z1, and let Z, = {0, 1,2, 3, 4,} (notation
as in Example 1). Show that the following bijection from Z, to Sis not an
isomorphism:
0→0 I-→2 2→4 3 →6 4→ 8.
5. Prove that the field R of real numbers is isomorphic to the ring of all 2 x 2
matrices of the form
,with a ER. [Hint: Consider the function f given
Courem
by f(a) =
6. Let Rand Sbe rings and let R be the subring of R × Sconsisting of all
elements of the form (a, Os). Show that the function f.R→R given by
f(a) = (a, Os) is an isomorphism.
7. Prove that Ris isomorphic to the ring S of all 2 × 2 matrices of the form
, where a ER.
8. Let Q(V2) be as in Exercise 39 of Section 3.1. Prove that the function
f:Q(V2) → Q(V2) given by f(a + bV2) = a – bV2 is an isomorphism.
9. If f:Z→Z is an isomorphism, prove that f is the identity map. [Hint: What
are f(1), f(1 + 1), ...?]
10. If R is a ring with identity and f:
ring S, prove that f(1r) is an idempotent in S. [Idempotents were defined in
Exercise 3 of Section 3.2.]
S is a homomorphism from R to a
Cape 2012 C Lang A Rig Rad May aot beopied aed od t ewle or in part Dee to deie d d per ey bepm neBodk edr . Btal ba
dd tg d d ny hate oa ingp Cn lning right dol c t da
3.3 Isomorphisms and Homomorphisms
81
11. State at least one reason why the given function is not a homomorphism.
(a) f:R→R and f(x) = Vx.
(b) g:E→E, where E is the ring of even integers and f(x) = 3x.
Expert Solution
Step 1
Let R and R' are rings.
A mapping is an isomorphism when
i)
ii)
iii) is injective
iv) is surjective
Define the mapping
by
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