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!
[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!
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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!
[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!
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Typo; that should be ae+bg.
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