Show that R={( a 0 / b c)} | a,b, c are elements of Z} i a ring with identity.

a 0 / b c is the matrix with a 0 the first row and b c the second row.

Question

Show that R={( a 0 / b c)} | a,b, c are elements of Z} i a ring with identity.

a 0 / b c is the matrix with a 0 the first row and b c the second row.

anonymous · Answer

also show that the map f:R -> Z given by f(a 0 / b c) = a is a surjective homomorphism.

What is the kernal of f?

terenzreignz · Answer

What does a 0 / b c 
mean?

anonymous · Answer

i didnt know how to make the matrix $$\left[\begin{matrix}a & 0 \ b&  c\end{matrix}\right]$$

terenzreignz · Answer

apparently, now you do :D

anonymous · Answer

well on the initial input math they dont give you the equation tools

terenzreignz · Answer

You have to memorise a few LaTeX tricks... but let's focus on your question, shall we? :)

anonymous · Answer

yes please

terenzreignz · Answer

You know how to multiply matrices?

anonymous · Answer

yes

terenzreignz · Answer

Well, then, for your first question, your only charge is pretty much to show closure... and show that matrix multiplication distributes over matrix addition... have you done this already?

anonymous · Answer

I can show closure but you'll have to elaborate on distributes over addition

terenzreignz · Answer

I mean, show that for 2x2 matrices A,B, and C
A(B + C) = AB + AC
(B + C)A = BA + BC

anonymous · Answer

oh show distribution okay

anonymous · Answer

that proves identity though?

terenzreignz · Answer

That's one of the ring-axioms, right? :D

anonymous · Answer

okay okay we are going to prove its a ring and then prove it has identity. I was going too far

terenzreignz · Answer

So you have it from here?

anonymous · Answer

except 
also show that the map f:R -> Z given by f(a 0 / b c) = a is a surjective homomorphism.

What is the kernal of f?

terenzreignz · Answer

Is it a homomorphism, though?
Show that first...

terenzreignz · Answer

show that for any two matrices A and B in R
f(A + B) = f(A) + f(B)
and
f(AB) = f(A)f(B)

That should be enough to show homomorphism-ness :D

anonymous · Answer

how do i determine the kernel?

terenzreignz · Answer

Kernel is simply the subset of R which is mapped to 0... namely, those 2x2 matrices which have 0 for their upper-left entry :)

anonymous · Answer

oh yeah thanks! I should be able to do it from here. I think I was just trying to over complicate it.

terenzreignz · Answer

No problem :)

anonymous · Answer

wait! how to I know its surjective? I mean to me it looks obvious that it is an onto function but  is there a formal way to show this?