Let Suppose that A α V B B 4 C A' B' C' is a commutative diagram of groups and that the rows are exact. Prove that (a) if and a are surjective, and ß is injective then y is injective. [If c E ker y, show there is a b E B with y(b) = c. Show that y' (B(b)) = 0 and deduce that 8(b) = for some d' E A'. Show there is an a E A with a (a) = d' and that B(v(a)) = B(b). Conclude that b = (a) and hence c = (a) c=y(b) = 0.] (b) if y', a, and y are injective, then ß is injective, if o, a, and y are surjective, then ß is surjective, af B is injective, a and y are surjective, then y is injective, eif ß is surjective, y and y' are injective, then a is surjective.
Let Suppose that A α V B B 4 C A' B' C' is a commutative diagram of groups and that the rows are exact. Prove that (a) if and a are surjective, and ß is injective then y is injective. [If c E ker y, show there is a b E B with y(b) = c. Show that y' (B(b)) = 0 and deduce that 8(b) = for some d' E A'. Show there is an a E A with a (a) = d' and that B(v(a)) = B(b). Conclude that b = (a) and hence c = (a) c=y(b) = 0.] (b) if y', a, and y are injective, then ß is injective, if o, a, and y are surjective, then ß is surjective, af B is injective, a and y are surjective, then y is injective, eif ß is surjective, y and y' are injective, then a is surjective.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let R be a ring with 1.
Suppose that
α
A
V
B
B
4
A'
B'
is a commutative diagram of groups and that the rows are exact. Prove that
(a) if p and a are surjective, and ß is injective then y is injective. [If c E ker y, show there
is a b E B with y(b) = c. Show that p' (B(b)) = 0 and deduce that B(b) = (a)
for some d' E A'. Show there is an a E A with a (a)= d' and that B((a)) = B(b).
Conclude that b = (a) and hence c = (b) = 0.]
(b) if y', a, and y are injective, then ß is injective,
if y, a, and y are surjective, then ß is surjective,
if B is injective, a and y are surjective, then y is injective,
eif ß is surjective, y and y' are injective, then a is surjective.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60aac33a-9c65-4f03-bee8-e937756cc77a%2F80535831-f224-49d6-a709-82148d9fed70%2Fscchfmi_processed.png&w=3840&q=75)
Transcribed Image Text:Let R be a ring with 1.
Suppose that
α
A
V
B
B
4
A'
B'
is a commutative diagram of groups and that the rows are exact. Prove that
(a) if p and a are surjective, and ß is injective then y is injective. [If c E ker y, show there
is a b E B with y(b) = c. Show that p' (B(b)) = 0 and deduce that B(b) = (a)
for some d' E A'. Show there is an a E A with a (a)= d' and that B((a)) = B(b).
Conclude that b = (a) and hence c = (b) = 0.]
(b) if y', a, and y are injective, then ß is injective,
if y, a, and y are surjective, then ß is surjective,
if B is injective, a and y are surjective, then y is injective,
eif ß is surjective, y and y' are injective, then a is surjective.
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