Can someone at least tell me what I should be looking for? Not only do I not know how to do these, but I do not even know what I should be looking for...

OK, so I get #1. I know what a maximal ideal is as well, but I don't get how I show that one exists for #3.

And 5 and 6 I don't really get, these are some of the more challenging questions we were asked. Any help there would be appreciated.