Question

@kirbykirby I need help understanding modern algebra concept. Please

Answer 1

Ok I can try

Answer 2

It says: Note every ideal in PID is trivial finitely generated. why? how it is trivial?

Answer 3

My problem: Let aR be a non-zero ideal in a PID R. Show that R/aR is a ring with only finitely many ideals.
What does it mean?
1) finitely many ideals. Does it mean R/aR has finite elements?

Answer 4

2) to show the whole thing, do I have to show R/aR is a ring before showing finite part?
Is it not that it is trivial? hahaha... since R is a PID, it is a ring for sure, right?
aR is a non-zero ideal, hence it is a ring also, right?
We have quotient R/aR is a ring also, right?(need prove or not?)

Answer 5

I am sorry for the new one: 3) Definition: An integral domain R is a PID if every ideal of R is principal.
That is, to me, if R is a PID , R is an integral domain + R is generated by an element + ideals of R are generated by their own element, right?
ha!! give me example, please.