Let R = { 0 a, b, d e Z} be the subring of M2 (Z) of upper triangular matrices. Define the map o : R → Z × Z by b = (a, d) a 0 d (1) Show that o is a homomorphism. (2) Show that o is onto. (3) Find ker(ø)

Let R = { 0 a, b, d e Z} be the subring of M2 (Z) of upper triangular matrices. Define the map o : R → Z × Z by b = (a, d) a 0 d (1) Show that o is a homomorphism. (2) Show that o is onto. (3) Find ker(ø)

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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Question

{[ ]
andez}:
a
Let R =
0 d
||a, b,
EZ} be the subring of M2 (Z) of upper triangular matrices.
Define the map ¢ : R → Z × Z by
• ([: :)-
a
= (a, d)
0 d
(1) Show that o is a homomorphism.
(2) Show that o is onto.
(3) Find ker(ø)
(4) Use the First Isomorphism Theorem to find what familiar ring R/ ker(o) is isomorphic to.

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