I am having a course in algebraic number theory.Dedekind domain plays an important role in number theory.It is defined as an integral domain all of whose fractional ideals are invertible.So we need to know about fractional ideals.Here is the definition:
Suppose $R$ is an integral domain and $K$ is the quotient field.Then a fractional ideal is a non-zero $R$-module $A\subset K$ such that there exists $r\in R\setminus\{0\}$ such that $rA\subset K$.
Can someone help me understand the reason behind it being called fractional ideal and give me some concrete examples of fractional ideals?Also can I characterize fractional ideals in terms of ideals of $R$?I also want to know how I should think of a fractional ideal,the picture is not very clear to me.
Your definition of a fractional ideal is a little bit wrong, but in a very significant way. You wrote
The last part of this definition is wrong. There's also no reason to exclude the zero-ideal from the definition. I'm not sure whether $\subset$ means strict inclusion to you or, so let's be clear that equality should be allowed in the definition too. Thus,
Now what do fractional ideals look like?
First note that if $I$ is an ideal of $R$ and $0 \not= r \in R$, then $I/r := \{k \in K \mid rk \in I \}$ is an $R$-submodule of $K$ and $rI \subseteq R$, so $I/r$ is a fractional ideal. The set $I/r$ consists of all the elements of $K$ which can be written as $a/r$ with $a \in I$.
On the other hand, if $A$ is a fractional ideal and $0 \not= r \in R$ is such that $rA \subseteq R$, then $rA$ is an $R$-submodule of $R$. The $R$-submodules of $R$ are precisely the ideals of $R$. Thus $rA = I$ for some ideal $I$ of $R$, and $A$ has the form $I/r$.
If a fractional element $k$ is of the form $a /r $ for some element $a \in R$ and $0 \not= r \in R$, then a fraction ideal $A$ is of the form $I/r$ for some ideal $I \subseteq R$ and $0 \not= r \in R$.