0
$\begingroup$

Let $M_n(R)$ be the matrix ring over a commutative ring $R$ and let $I$ be an ideal of $R$.

1) Show that if $R=F$ is a field then the only nonzero ideal of $M_n(F)$ is $M_n(F)$ itself.

2) Let $M_n(I)$ be the subset of $M_n(R)$ consisting of matrices with entries in the ideal $I$. Show that $M_n(I)$ is an ideal of $M_n(R)$ and describe the quotient ring $M_n(R)/M_n(I)$ by a matrix ring.

3) Show that if $I$ is a maximal ideal then $M_n(I)$ is a maximal ideal of $M_n(R)$.


Any advice on how to do these questions would be greatly appreciated!

$\endgroup$
2
$\begingroup$

1) See Why is the ring of matrices over a field simple? for this.

2) By the way matrix multiplication works, $M_n(R)M_n(I)\subseteq M_n(R)$. One might guess that the quotient is $M_n(R/I)$, so lets look at it backwards: What is the kernel of the canonical homomorphism $M_n(R)\to M_n(R/I)$? (Note that this latter question at the same time shows that $M_n(I)$ is an ideal and what th equotient is)

3) Follows from 1 and 2 as $R/I$ is a field.

$\endgroup$
  • $\begingroup$ @egreg Oh, right.I need to combine row and column operations and linear combine results. Your link gives a much better answer. $\endgroup$ – Hagen von Eitzen May 26 '13 at 16:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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