Let $R$ be an infinite commutative ring. Which of following options is false?
Center of $M_2(R×R)$ is nontrivial.
$ M_2(R×R) \cong M_2(R)×M_2(R)$
The number of units in $M_2(R ×R)$ is infinite.
The number of two-sided ideals in $M_2(R ×R)$ maybe are finite.
Now "1" is true because every diagonal matrix is in center of this ring.
"4" is true because :
Theorem: every two-sided ideal of $M_n(R)$ has the form $M_n(I)$ for some unique two-sided ideal $I$ of $R$.