Actually depending on the context of the sentence these structures are actually interchangeable.
People can commit murder. - People can commit crime.
I stopped him before he could commit a murder. - I stopped him before he could commit a crime.
The one without an article is referring to the action itself. Murder/Rape/Suicide/Adultery are all representative of their actions. a good example is "are you able to commit murder?" this is asking if you have the ability to do murder. While, "are you able to commit a murder?" is asking more specifically if you are able to murder within some indefinite case. So it is not if you are able to murder as many can technically murder someone, it is asking if you can actually go through and commit and do the actions of a murder. Putting "the" in front instead of 'a' makes it more clear as it is asking, if you can commit the actions of The Murder, a specific defined murder while with 'a' it is an undefined murder.
For the case of crime it is the same principle, just in reverse. With an article adjective it is asking if you can commit the actions of a/the certain crime. While without it it is asking if you can commit crime as in if it is possible.
Now Both of these examples were with the sentence "are you able to..." but these are general connotations, that can be applied to any sentence. The problem that makes your issue confusing is that you don't have a sentence around it; it is simply a fragment of a sentence.So this will only serve to confuse you more. A good habit to think about is whenever you are given these confusing fragments try putting them in a sentence and switching the rules between them and see if it still makes sense.