35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and Z O RXRXRXRand M(R) (4) Qand R O Z X Z, and Z
35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and Z O RXRXRXRand M(R) (4) Qand R O Z X Z, and Z
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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#35 from the Textbook
![b MATLAB: An Introduction with A X
Thomas W. Hungerford - Abstrac ×
+
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
-- A' Read aloud V Draw
F Highlight
O Erase
105
of 621
30. Let f:R→S be a homomorphism of rings and let K = {rɛR|f(F) = 0s}.
Prove that K is a subring of R.
31. Let f:R →Sbe a homomorphism of rings and Ta subring of S.
Let P = {rER|f(r) ET}. Prove that P is a subring of R.
32. Assume n = 1 (mod m). Show that the function f:Z,→Zm given by
f([xL.) = [nx] is an injective homomorphism but not an isomorphism when
n2 2 (notation as in Exercise 12(e)).
33. (a) Let T'be the ring of functions from R to R, as in Example 8 of Section 3.1.
R be the function defined by 0(f) =f(5). Prove that 0 is a
Let 0:
surjective homomorphism. Is 0 an isomorphism?
(b) Is part (a) true if 5 is replaced by any constant ceR?
34. If f: →S is an isomomorphism of rings, which of the following properties
are preserved by this isomorphism? Justify your answers.
(a) aERis a zero divisor.
Carite 2012 0 la A Righ Ra May aot be opind cemtr daplicd a whae or in part D to emis d. rd perty com y bepmd te eBet deCh . Bal w t
ded thatny ppd doot dany t he o i e C lamtng erigbto dnol cot eta i
3.3 Isomorphisms and Homomorphisms
83
(b) aER is idempotent.*
(c) Ris an integral domain.
35. Show that the first ring is not isomorphic to the second.
(b) RXRXR x R and M(R)
(a) E and Z
(c) Z, × Zu and Z6
(e) ZXZ, and Z
36. (а) If f:
NĒZ, f(nr) = nf(r).
(b) Prove that isomorphic rings with identity have the same characteristic.
[See Exercises 41–43 of Section 3.2.]
(d) Q and R
() Z, × Z, and Z16
S is a homomorphism of rings, show that for any re R and
(c) If f:
S have the same characteristic?
S is a homomorphism of rings with identity, is it true that R and
37. (a) Assume that e is a nonzero idempotent in a ring R and that e is not a zero
divisor.* Prove that e is the identity element of R. [Hint: = e (Why?). If
aER, multiply both sides of e? - e by a.]
(b) Let S be a ring with identity and Ta ring with no zero divisors. Assume
that f: →Tis a nonzero homomorphism of rings (meaning that at least
one element of S is not mapped to 07). Prove that f(1) is the identity
element of T. [Hint: Show that f(ls) satisfies the hypotheses of part (a).]
11:05 AM
e Type here to search
EPIC
Ai
EPIC
99+
10/30/2020](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2Fa1e93bd9-b323-47c5-8c90-2c394a8aa581%2F5em4dzj_processed.png&w=3840&q=75)
Transcribed Image Text:b MATLAB: An Introduction with A X
Thomas W. Hungerford - Abstrac ×
+
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
-- A' Read aloud V Draw
F Highlight
O Erase
105
of 621
30. Let f:R→S be a homomorphism of rings and let K = {rɛR|f(F) = 0s}.
Prove that K is a subring of R.
31. Let f:R →Sbe a homomorphism of rings and Ta subring of S.
Let P = {rER|f(r) ET}. Prove that P is a subring of R.
32. Assume n = 1 (mod m). Show that the function f:Z,→Zm given by
f([xL.) = [nx] is an injective homomorphism but not an isomorphism when
n2 2 (notation as in Exercise 12(e)).
33. (a) Let T'be the ring of functions from R to R, as in Example 8 of Section 3.1.
R be the function defined by 0(f) =f(5). Prove that 0 is a
Let 0:
surjective homomorphism. Is 0 an isomorphism?
(b) Is part (a) true if 5 is replaced by any constant ceR?
34. If f: →S is an isomomorphism of rings, which of the following properties
are preserved by this isomorphism? Justify your answers.
(a) aERis a zero divisor.
Carite 2012 0 la A Righ Ra May aot be opind cemtr daplicd a whae or in part D to emis d. rd perty com y bepmd te eBet deCh . Bal w t
ded thatny ppd doot dany t he o i e C lamtng erigbto dnol cot eta i
3.3 Isomorphisms and Homomorphisms
83
(b) aER is idempotent.*
(c) Ris an integral domain.
35. Show that the first ring is not isomorphic to the second.
(b) RXRXR x R and M(R)
(a) E and Z
(c) Z, × Zu and Z6
(e) ZXZ, and Z
36. (а) If f:
NĒZ, f(nr) = nf(r).
(b) Prove that isomorphic rings with identity have the same characteristic.
[See Exercises 41–43 of Section 3.2.]
(d) Q and R
() Z, × Z, and Z16
S is a homomorphism of rings, show that for any re R and
(c) If f:
S have the same characteristic?
S is a homomorphism of rings with identity, is it true that R and
37. (a) Assume that e is a nonzero idempotent in a ring R and that e is not a zero
divisor.* Prove that e is the identity element of R. [Hint: = e (Why?). If
aER, multiply both sides of e? - e by a.]
(b) Let S be a ring with identity and Ta ring with no zero divisors. Assume
that f: →Tis a nonzero homomorphism of rings (meaning that at least
one element of S is not mapped to 07). Prove that f(1) is the identity
element of T. [Hint: Show that f(ls) satisfies the hypotheses of part (a).]
11:05 AM
e Type here to search
EPIC
Ai
EPIC
99+
10/30/2020
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