1. ## Rings

Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

a) Show that S is a subring of M(R).
b) Show that S is not a field.

I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication. What does the normally defined addition and multiplication statement mean?

2. Originally Posted by empressA
Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

a) Show that S is a subring of M(R).
b) Show that S is not a field.

I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication.
No, a subring must also be closed under multiplication. Saying that it is a ring rather than a field means that there may be elements other than 0 which do not have multiplicative inverses.

What does the normally defined addition and multiplication statement mean?
The nddition addition and multiplication of matrices, of course.

$\displaystyle \begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}+ \begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix}= \begin{bmatrix}0 & r+s \\ 0 & 0\end{bmatrix}$

$\displaystyle \begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix}$ = ?

3. ok...but when i multiply those matrices i end up with [0 0, 0 0]...would that still make it closed under multiplication..i know that it satisifes all the other requirements for a ring but am not sure if its closed under addition. Thats what I meant in the beginning I worded it wrong