Let $p$ be a prime number and $n$ be a natural number. Does $M\_n({\mathbf{Z}}\_p)$, the ring of all $n\times n$ matrices over the field ${\mathbf{Z}}\_p$ form a Baer\*-ring? 


If not, what about for some special $p$ and $n>2$?



Is $M_n(Z_p)$ a Baer*-ring?
If $R$ is a ring such that $M\_n R$ is Baer\*, then all sums of the form $\sum\_{i=1}^n r\_i r\_i^\*$ must be nonzero, unless all $r\_i = 0$. With \*-transpose involution on matrix rings over $R$, this excludes $p \equiv 1 \pmod 4$ and $n \geq 2$, as well as $p\equiv -1 \pmod 4$ and $n \geq 3$, whenever the characteristic of $R$ is $p$. 


