I'm trying to find a book about ring theory such that
- a ring doesn't need to be commutative and to have a multiplicative identity
- $A:=\left\{\begin{pmatrix}a&0\\0&0\end{pmatrix}:a\in \mathbb{R}\right\}$ is considered as a subring of the ring of matrices $M_{2,2}(\mathbb{R})$.
A notion of subring such that the above item is true is necessary in the following proposition: let $R$ be a ring with unity $1_R$. If $A\subseteq R$ is a nontrivial subring with unit $1_A$, then either $1_A=1_R$ or $1_A$ is a zero divisor in $R$.
It would be nice, but not necessary, if the book also covers topics such as modules and homological algebra.