Abstract Algebra Matrix Proof?
Herstein's abstract algebra: Chapter 4 section 3 number 19
If R = {( a b ) | a, b, c real} and I = {( 0 b ) | b real }
.........{( 0 c ) |.................}..............{( 0 0 ) |...........}
values in between the {} are matrices.
Show that:
(a) R is a ring
(b) I is an ideal of R
(c) R/I ≈ F ⊕ F, where F is the field of real numbers
Comments
(a) It suffices to show that R is a subring of M₂(R).
Since the difference and products of upper triangular matrices are again upper triangular, this is indeed the case.
For specifically, let
A = (a b)
......(0 c), and
B = (d e)
......(0 f) be in R.
Then, A - B =
(a-d b-e)
(0 c-f) is also in R.
AB =
(ad ae+bf)
(.0......cf...) is in R.
------------------
b) For A = (0 a; 0 0), B = (0 b; 0 0) in I,
A + B = (0 a+b; 0 0) is in R.
For A = (0 a; 0 0) and I, and C = (b c; 0 d) in R:
AC =
(0 a)(b c)...(0 ac)
(0 0)(0 d).=(0 0) is in R, and
CA =
(b c)(0 a)...(0 ab)
(0 d)(0 0).=(0 0) is in R.
Hence, I is an (two-sided) ideal of R.
---------------
(c) Consider the map F : R → F ⊕ F defined by F((a b; 0 c)) = (a, c).
It's easy to see that this is a surjective ring homomorphism,
with ker F = {(0 b; 0 0) : b in F} = I.
So, the result follows from the first isomorphism theorem.
I hope this helps!
Yeah...i think of you may! I advise in case you may shop up with doing each and all of the paintings it is handy, and you nevertheless get A's in college...it is going to be no undertaking for you. additionally, i'm thinking of majoring in Maths additionally...while that's time