I am considering rings of the form $$ \begin{pmatrix} M_{11} & \cdots & M_{1n} \\ \vdots & & \vdots \\ M_{n1} & \cdots & M_{nn}\end{pmatrix}$$ with $M_{ij}$ being either just $0$ or some fixed field (or ring) $F$ (I will call them "matrix-style"). Some of these constructions actually are a ring (not necessarily with a $1$) by elementwise addition and matrix-multiplication, e.g. $$ \begin{pmatrix} 0 & 0 & 0 \\ F & F & 0 \\ F & 0 & F \end{pmatrix}$$ is a ring and $$ \begin{pmatrix} 0 & 0 & F \\ 0 & F & 0 \\ F & 0 & F \end{pmatrix}$$ is not a ring. It is quite easy to see if such a construction is a ring by multiplying two general matrices and checking if it is still of this form. However, I wonder if there is an easier way to see if such a construction is a ring without having to check manually? Conversely is there a criterion which shows that certain constructions cannot be rings? (I have checked all possible $3\times 3$ "matrix-style" rings on the computer and I do not see any patterns.)
More generally given such a "matrix-style" ring, is it possible to describe "matrix-style" (left/right) ideals of this ring (of course one could again check this manually)?