Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Show that $Z(M_n(R))$ consist of diagonal matrices with entries $Z(R)$.

share|improve this question
We do not like taking orders. Instead, please indicate how you came across this problem; why it interests you; what you already know about it, and about the concepts mentioned in it; how far you got in your efforts to solve it; where you got stuck; and so on, and so on. Engage with us, so we can engage with you. –  Gerry Myerson Jan 22 '13 at 6:32
Since you are new to this site, please consider reading this: How to ask a homework question?. In particular, you should use homework tag if your question comes from a homework. I wrote this comment because the question sounds homework-like. –  Julian Kuelshammer Jan 22 '13 at 6:34
Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. –  Julian Kuelshammer Jan 22 '13 at 6:35
For some basic information about writing math at this site see e.g. here, here, here and here. –  Julian Kuelshammer Jan 22 '13 at 6:37

1 Answer 1

Here are some things to think about which should put you in the right direction.

Suppose $A=(a_{ij})\in Z(M_n(R))$. Let $E_{ij}$ be the matrix whose $i,j$ entry is $1$, and all other entries are $0$. Then the equations $$ E_{ii}A=AE_{ii} $$ for $1\leq i\leq n$ implies that $A$ is necessarily diagonal. (Why?) Furthermore, $$ AE_{ij}=E_{ij}A $$ for $1\leq i,j\leq n$ implies that $a_{ii}=a_{jj}$ for all $i$ and $j$. (Why?) Hence $A=aI_n$ for some $a\in R$. But notice that $$ aI_n(bI_n)=bI_n(aI_n),\quad \forall b\in R $$ implies that $a\in Z(R)$.

share|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.