Consider the set S of matrices of the form
\left[\begin{matrix} a & b\ 0 & c\end{matrix} ight]
with a, b, c real numbers.

a. show that S is a ring with operations given by addition and multiplication of matrices. Is S a commutative ring?
b. Describe the units of the ring S.
c.Is S a surbring of the ring M_{2x2}(R) of 2x2 matrices? Is it ideal of M_{2x2}(R)

Question

Consider the set S of matrices of the form 
\left[\begin{matrix} a & b\ 0 & c\end{matrix}ight]
 with a, b, c real numbers.

a. show that S is a ring with operations given by addition and multiplication of matrices. Is S a commutative ring?
b. Describe the units of the ring S.
c.Is S a surbring of the ring M_{2x2}(R) of 2x2 matrices? Is it ideal of M_{2x2}(R)

amistre64 · Answer

what condition are required to define a ring?  i recall its a group, plus, a(x+y) = ax+ay

anonymous · Answer

I can do a and prove the axioms, but I don't understand b & c.

anonymous · Answer

an abelian group under addition, which it is

anonymous · Answer

multiplication needs to be associative, which is is because it is for matrices in general

anonymous · Answer

i guess since the set of two by two matrices forms a ring in any case, what you really need to show is that it is closed

anonymous · Answer

units would be two by two matrices with inverses, meaning the determinant is not zero, meaning in this case $ac\neq 0$ and so $a\neq 0$ and $c\neq 0$

anonymous · Answer

is it an ideal?

anonymous · Answer

no its not