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.

I was stuck with the seemingly simple homework problem:

A $R$-module is injective if and only if every exact sequence $$0\rightarrow E\rightarrow B\rightarrow R/I\rightarrow 0$$ splits.Here $I$ is an ideal of $R$.

The only if direction is straightforward since if $E$ is injective, then every exact sequence of former type must splits. But I do not know how to prove $E$ is injective given the above condition. The natural direction is to use Baer's criterion. However the above exact sequence seems to be quite difficult to apply this criterion. So I am stuck.

share|improve this question

1 Answer 1

up vote 4 down vote accepted

Suppose $I\to R$ is the inclusion map from an left ideal of $R$ and that $f:I\to E$ is a map. We have a diagram

0  --> I --> R --> R/I --> 0
       |
       |
       V
       E

So we can construct the push-out, to get a diagram with exact rows

0 --> I --> R --> R/I --> 0
      |     |      |
      |     |      |
      V     V      V
0 --> E --> ? --> R/I ---> 0

where ? is some module. Now use the hypothesis on the bottom row.

share|improve this answer
    
Thank you! This is really clear. –  Bombyx mori Dec 13 '12 at 21:47
    
Can I ask how do I get the arrow from ? to R/I? –  Bombyx mori Dec 13 '12 at 21:58
    
It comes out from the construction of push-outs. –  Mariano Suárez-Alvarez Dec 13 '12 at 22:14
    
I see. Thanks!! –  Bombyx mori Dec 14 '12 at 0:38

Your Answer

 
discard

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.