(January 2012) Let \[ R=\left\{\begin{pmatrix}a&b\\0&c\end{pmatrix}:a,b,c\in\mathbb{R}\right\}. \] You may assume that \(R\) is a ring under the operations of matrix addition and matrix multiplication. Prove that the set \[ I=\left\{\begin{pmatrix}a&b\\0&0\end{pmatrix}:a,b\in\mathbb{R}\right\} \] is an ideal of \(R\).
Solution
Proof. (I non-empty) [ 0 0 0 0] ∈I, so I⁄= ∅. (I closed under−) Given r = [a b 0 0] and s = [c d 0 0] , r−s = [a−c b −d 0 0] ∈I. (Sticky) Given r = [a b 0 c] ∈ R and s = [x y 0 0] , rs = [ax ay 0 0] ∈ I and sr = [ax bx +cy 0 0] ∈ I. QED Here is problem that uses a definition you are not expected to have seen before.