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(i) morphisms 3. State what it means for a sequence of left R-module M N P Q to be "exact". (ii) Let f: Z²Z² and g: Z² → Z be the morphisms given by f(a, b) = (2a + 3b, 6a+ 9b) and Show that the sequence is exact. g(x, y) = y-3x. z²Z² ZO (iii) State what is meant by a "short" exact sequence, and what it means for a short exact sequence to be "split".

(i) morphisms 3. State what it means for a sequence of left R-module M N P Q to be "exact". (ii) Let f: Z²Z² and g: Z² → Z be the morphisms given by f(a, b) = (2a + 3b, 6a+ 9b) and Show that the sequence is exact. g(x, y) = y-3x. z²Z² ZO (iii) State what is meant by a "short" exact sequence, and what it means for a short exact sequence to be "split".

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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(i) morphisms 3. State what it means for a sequence of left R-module M 4 N , P Q to be "exact". Let f: Z? → Z² and g: Z? → Z be the morphisms (ii) given by f (a, b) = (2a + 3b, 6a + 9b) and g(x, y) = y – 3x. Show that the sequence Z² , z? 9+ Z → 0 is exact. (iii) State what is meant by a “short" exact sequence, and what it means for a short exact sequence to be “split".
Transcribed Image Text:(i) morphisms 3. State what it means for a sequence of left R-module M 4 N , P Q to be "exact". Let f: Z? → Z² and g: Z? → Z be the morphisms (ii) given by f (a, b) = (2a + 3b, 6a + 9b) and g(x, y) = y – 3x. Show that the sequence Z² , z? 9+ Z → 0 is exact. (iii) State what is meant by a “short" exact sequence, and what it means for a short exact sequence to be “split".
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