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.

Let $A,B$ be $n\times n$ matrices with entires in a PID $D$ and $\det AB\neq 0$. Suppose diag$\{a_i\}$, diag$\{b_i\}$, and diag$\{c_i\}$ are normal froms for $A$, $B$, and $AB$. In particular, $a_i|a_{i+1}$. Why must $a_i$ and $b_i$ both divide $c_i$?

I think this requires an application of the structure theorem for modules over a PID, possibly viewing $A$ and $B$ as relations matrices, but I'm not sure how to solve the problem. Any help would be appreciated.

(I am studying for a test -- this is not an exam or homework problem. It is number 5 on page 194 of Jacobson's Basic Algebra I.)

share|improve this question
add comment

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.