subring of 2x2 matrices with real entries. Let $S = \left\{\begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix} :a,b \in \mathbb {R} \right\}$. Prove that the set $S$ is a subring of $M_2 (R)$ with its usual addition and multiplication. Is $S$ a ring with identity? Is it a commutative ring? Describe the units of this ring.


yes this is a subring of $M_2 (R)$, $S$ has an identity but its not the identity in $M_2 (R)$ it is commutative and it has no units. 
Does the fact that the identity is not the same matter?
$ \begin{bmatrix}1&amp;0\\0&amp;0\end{bmatrix} \in S$ so S is non empty.

$ \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix} \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix}= \begin{bmatrix}ac+0&amp;ad+0\\0&amp;0\end{bmatrix}$ $ac,ad \in \mathbb {R}$ closed under multiplication

$ \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix} + \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix} = \begin{bmatrix}a+c&amp;b+d\\0&amp;0\end{bmatrix}$ $(a+c),(b+d) \in \mathbb {R}$ closed under addition

$ \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix} \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix}= \begin{bmatrix}ac+0&amp;ad+0\\0&amp;0\end{bmatrix}$

$ \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix} \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix}= \begin{bmatrix}ca+0&amp;cb+0\\0&amp;0\end{bmatrix}$

so not a commutative ring.

$\begin{bmatrix}1&amp;0\\0&amp;1\end{bmatrix} $ is not in S so S has no unity.

Since $ \forall s \in S $ the det(s)=0 nothing is a unit.
$ \begin{bmatrix}1&amp;0\\0&amp;0\end{bmatrix} \in S$ so S is non empty.

$ \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix} \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix}= \begin{bmatrix}ac+0&amp;ad+0\\0&amp;0\end{bmatrix}$ $ac,ad \in \mathbb {R}$ closed under multiplication

$ \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix} + \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix} = \begin{bmatrix}a+c&amp;b+d\\0&amp;0\end{bmatrix}$ $(a+c),(b+d) \in \mathbb {R}$ closed under addition

$ \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix} \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix}= \begin{bmatrix}ac+0&amp;ad+0\\0&amp;0\end{bmatrix}$

$ \begin{bmatrix}c&amp;d\\0&amp;0\end{bmatrix} \begin{bmatrix}a&amp;b\\0&amp;0\end{bmatrix}= \begin{bmatrix}ca+0&amp;cb+0\\0&amp;0\end{bmatrix}$

so not a commutative ring.

$\begin{bmatrix}1&amp;0\\0&amp;1\end{bmatrix} $ is not in S so S has no unity.

Since $ \forall s \in S $ the det(s)=0 nothing is a unit.