the identity map] 30. Let f:R+ S be a homomorphism of rings and let K- {reR S() - Os). Prove that K is a subring of R. 31. Let f:R+S be a homomorphism of rings and Ta subring of S. Let P= frER|f(r)ET}. Prove that Pis a subring of R. 32. Assume n=1 (mod m). Show that the function f:Z Zm given by S(xL) = [nxl 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.
the identity map] 30. Let f:R+ S be a homomorphism of rings and let K- {reR S() - Os). Prove that K is a subring of R. 31. Let f:R+S be a homomorphism of rings and Ta subring of S. Let P= frER|f(r)ET}. Prove that Pis a subring of R. 32. Assume n=1 (mod m). Show that the function f:Z Zm given by S(xL) = [nxl 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.
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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#31 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
104
of 621
(ac, bc). Show that R is a ring.
(b) Show that the ring of part (a) is isomorphic to the ring of all matrices in
M(R) of the form
Coursesmait
25. Let L be the ring of all matrices in M(Z) of the form
Show that the
function f:
not an isomorphism.
Z given by
= a is a surjective homomorphism but
26. Show that the homomorphism g in Example 7 is injective but not surjective.
27. (a) If g:
Sand f.S →Tare homomorphisms, show that f° g:
Tis a
homomorphism.
(b) If fand g are isomorphisms, show that fog is also an isomorphism.
28. (a) Give an example of a homomorphism f:R -→S such that R has an identity
but S does not. Does this contradict part (4) of Theorem 3.10?
(b) Give an example of a homomorphism f:R→S such that Shas an identity
but Rdoes not.
S be an isomorphism of rings and let g:
R be the inverse
29. Let f:
function of f (as defined in Appendix B). Show that g is also an isomorphism.
[Hìnt: To show g(a + b) = g(a) + g(b), consider the images of the left- and
right-hand side under fand use the facts that fis a homomorphism and f•g is
the identity map]
30. Let f:R→S be a homomorphism of rings and let K = {rɛR|f(r) = 0s}.
Prove that K is a subring of R.
31. Let f:R→S be a homomorphism of rings and Ta subring of S.
Let P = {rER|f)ET}. Prove that P is a subring of R.
32. Assume n = 1 (mod m). Show that the function f:Z,→ Zm given by
S([xL.) = [nx] is an injective homomorphism but not an isomorphism when
nz 2 (notation as in Exercise 12(e)).
33. (a) Let Tbe 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: →Sis an isomomorphism of rings, which of the following properties
are preserved by this isomorphism? Justify your answers.
(a) aERis a zero divisor.
Carit 2012 C la ARigha Ra May aot be opind candrdaplicdn whale or part Dtodmis d. rd perty cod y bepd teeBoat dte . Balvw t
ded tat oy appndco dootdany et the o gpet C lamog mr rigbto ve ddol cotttefa ghta t ire
3.3 Isomorphisms and Homomorphisms
83
11:03 AM
e Type here to search
口
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%2Fb0a335cc-b2c0-4274-9fa7-c6728db28819%2Ffmt75ol_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
104
of 621
(ac, bc). Show that R is a ring.
(b) Show that the ring of part (a) is isomorphic to the ring of all matrices in
M(R) of the form
Coursesmait
25. Let L be the ring of all matrices in M(Z) of the form
Show that the
function f:
not an isomorphism.
Z given by
= a is a surjective homomorphism but
26. Show that the homomorphism g in Example 7 is injective but not surjective.
27. (a) If g:
Sand f.S →Tare homomorphisms, show that f° g:
Tis a
homomorphism.
(b) If fand g are isomorphisms, show that fog is also an isomorphism.
28. (a) Give an example of a homomorphism f:R -→S such that R has an identity
but S does not. Does this contradict part (4) of Theorem 3.10?
(b) Give an example of a homomorphism f:R→S such that Shas an identity
but Rdoes not.
S be an isomorphism of rings and let g:
R be the inverse
29. Let f:
function of f (as defined in Appendix B). Show that g is also an isomorphism.
[Hìnt: To show g(a + b) = g(a) + g(b), consider the images of the left- and
right-hand side under fand use the facts that fis a homomorphism and f•g is
the identity map]
30. Let f:R→S be a homomorphism of rings and let K = {rɛR|f(r) = 0s}.
Prove that K is a subring of R.
31. Let f:R→S be a homomorphism of rings and Ta subring of S.
Let P = {rER|f)ET}. Prove that P is a subring of R.
32. Assume n = 1 (mod m). Show that the function f:Z,→ Zm given by
S([xL.) = [nx] is an injective homomorphism but not an isomorphism when
nz 2 (notation as in Exercise 12(e)).
33. (a) Let Tbe 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: →Sis an isomomorphism of rings, which of the following properties
are preserved by this isomorphism? Justify your answers.
(a) aERis a zero divisor.
Carit 2012 C la ARigha Ra May aot be opind candrdaplicdn whale or part Dtodmis d. rd perty cod y bepd teeBoat dte . Balvw t
ded tat oy appndco dootdany et the o gpet C lamog mr rigbto ve ddol cotttefa ghta t ire
3.3 Isomorphisms and Homomorphisms
83
11:03 AM
e Type here to search
口
EPIC
Ai
EPIC
99+
10/30/2020
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