I don’t think there is a rule for 間 (かん) o.o
So, I had another look at the sentence you had a problem with until I understood it. Took a few minutes. It’s something I rarely encounter/use in my Japanese course xD’
I’m not going to explain in too much detail, you could probably fill pages with this…
First of all いる and ある
Well, you aren’t really wrong (although it’s not he/she DID something but more he/she IS). They are both verbs commonly used to describe that someone/something exists
- for humans/animals it’s いる: to be (of animate objects), to exist (かれしがいる – I have a boyfriend; ねこがいる – There is a cat)
- for objects it’s ある: to be (usu. of inanimate objects), to exist, to have (ほんがある – I have a book)
Of course there are other uses for あるand いる.
For example in the first sentence you posted, you can find the ~ている form of verbs. The first sentence you posted was わたしは いっかげつかん にほんごを べんきょう しています. The べんきょう していますused here is the ている form of べんきょうする (to study). The ている form is used for continuous actions (like the English continuous tense).
Now the last sentence you asked about:
I wouldn’t really call this a question between いる and ある. This is a question of negation.
Let’s look at nouns first.
だいがくせいです。I am a university student
- だいがくせいじゃないです I am not a university student.
- is the form you are probably already familiar with. But you can also negate it with ではありません (or で(は)ない in informal speech)
- だいがくせいではありません。 I am not a university student.
It’s the same with adjectives. Let’s use さむいas example: さむいです。It’s cold
Let’s negate this then:
- さむくないです。 It’s not cold
- さむくありません。It’s not cold
The sentence かれは さむく ありませんでした is the same (only in past tense).
As for the difference between the ない and ありません, it’s a matter of politeness.
So I wrote all this and then I find a page where this is explained. Oh well, have a look: http://www.guidetojapanese.org/learn/grammar/polite
(If you don’t want to read everything, look at the last third of the page)
Does this more or less answer your question? ^^