Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let n be arbitrary positive integer and R arbitrary ring (perhaps, non-associative). Let's denote $M_n(R)$ the set of all $n х n$ matrices with entries from R.

As i know if R is non-trivial commutative ring without zero divisors then scalar matrices are the only matrices which commutate with all other matrices over R.

By scalar matrix i mean diagonal matrix such that all its diagonal entries are equal.

I have the following question: what if R is non-trivial ring WITH zero divisors? Is it possible that there is matrix $M \in M_n(R)$ such that $\forall A\in M_n(R) ~~ AM = MA$ but M is non-diagonal or at least non-scalar.

Thanks in advance.

share|cite|improve this question
up vote 3 down vote accepted

$M$ is in the center of $M_n(R)$ iff $M$ commutes with all matrices $E_{ij}$ (the $(i,j)$th entry equals $1$ and others $0$). From here you can easy obtain the answer.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.