determinant of a $2\times2$ matrix, sufficiency of inverse What is the simplest example of a nontrivial ring in which these two conditions are not equivalent for any $2\times 2$ matrix $A$:
(1) there is a $2\times2$ matrix $B$ with $A\cdot B=1$
(2) $a_{11}a_{22}\neq a_{21}a_{21}$
To make the question more interesting: can the structures such that (1) is (not) equivalent to (2) be characterized?
 A: Condition (2) as written is not equivalent to invertibility of a matrix over any (commutative) ring and it should be
$$
a_{11}a_{22}\ne a_{12}a_{21}
$$
A truly equivalent condition is that the determinant is invertible in the ring, so it shouldn't be really difficult to find an example over the simplest ring which is not a field.
A: Try $\begin{pmatrix}1 & 1\\ 0 & 2\end{pmatrix}\in M_2(\mathbb{Z}_4)$. (2) doesn't imply (1) for the reason that $2$ is no invertible. Just as mentioned before by @egreg
A: Hint:
for 
$$
a_{11}a_{22}\ne a_{12}a_{21}
$$
consider the subring of $M_2(\mathbb{R})$ with matrices of the form:
$$
\begin{bmatrix}
\mathbb{Z}&\mathbb{Q}\\
0&\mathbb{Z}
\end{bmatrix}
$$
Many matrices with non null determinant are not invertible in this ring.
A: A matrix $A\in M_2(R)$, where $R$ is a ring, is invertible iff $\det A\in U(R)$, where $U(R)$ is the set of the units of the ring $R$.
So a simple example is given by the ring $\Bbb Z$, any matrices whose determinant is different to $\pm 1$ is not invertible.

Example 
  $$ A=
\begin{pmatrix}
2 & 1\\
3 & 4
\end{pmatrix}
$$
$\det A=5$ so $A$ is not invertible in $M_2(\Bbb Z)$, even if is invertible in $M_2(\Bbb R)$. 

