0

Let $R$ be a ring. The center of $R$ is the set $C(R) = \{c ∈ R : cr = rc, ∀r ∈ R\}$.

Determine elements in the center of the $n × n$ matrix ring $M_n (R) $ for any $n ≥ 2$.

So, we have that center $C(R)=\{x \in R / cr=rc, \forall r \in R\}$

and we need to find $C(M_n(R))\,\,\,\,\, (n \ge 2)$

The only thing that I could think of is to start looking $C(M_2 (R))$, taking $M_2(R)$ for $\begin{bmatrix}a&b\\c&d\end{bmatrix} \ge 0$ and if $a, b, c, d \ge 0$ in $R$ then, $I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix} \ge 0$ but then $\begin{bmatrix}1&-1\\0&2\end{bmatrix} \not\ge 0$

user26857
  • 52,094
cele
  • 2,247
  • I do not see how it is a duplicate. I have looked over the other problem and it does not seem the same to me – cele Apr 23 '15 at 00:01
  • This is unclear; Is $R$ an ordered ring? What relevance does the ordering have to finding the center of $M_n(R)$? Assuming you meant $\mathbb R$, which is very confusing because you are using $R$ also to denote a generic ring, it still is very unclear what the stuff about positivity in the last sentence has to do with finding centers. – Jonas Meyer Apr 23 '15 at 03:03

0 Answers0