Proving a ring homorphism
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Proving a ring homorphism

[From: ] [author: ] [Date: 11-04-22] [Hit: ]
cf+dh]; this has image under the map equal to ae = bg.Needless to say ae and ae+bg are generally not equal to one another.I hope this helps!-Typo; that should be ae+bg.......
Consider the mapping from M2(Z) into Z given by the matrix
[a b]
[c d] --> a.

Prove or disprove that this is a ring homomorphism.

I know that it doesn't hold for Multiplication(so it isn't a ring homomorphism) but i would really love how the answer was gotten. Show addition and Multiplication, if you could.
Thanks!

-
Take two matrices in M2(Z)
[a b]..[e f]
[c d],.[g h]

These matrices map to a and e, respectively; to the product of the images is ae.

On the other hand, multiplying the matrices first yields
[ae+bg...af+bh]
[ce+dg...cf+dh]; this has image under the map equal to ae = bg.

Needless to say ae and ae+bg are generally not equal to one another.

I hope this helps!

-
Typo; that should be ae+bg.

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keywords: Proving,homorphism,ring,Proving a ring homorphism
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