Let H be an R module contained in F, and assume, for some d in R, d*H lies entirely in R. In other words H has a common denominator d, and when you multiply through by d you get an ideal in R. In this case H is called a "fractional ideal".
If R is the integers and F is the rationals, the integers and half integers form a Z module of dimension 2 that is also a fractional ideal with common denominator 2. In contrast, H could be the set of rational numbers whose denominators are powers of 2. This is a Z module that is not a fractional ideal.
If H is contained in R it is a fractional ideal by default, hence every ideal is a fractional ideal.
If H is finitely generated, let d be the product of the denominators of the generators. Now d drives the generators, and all of H, into R. Therefore every finitely generated R module contained in F is a fractional ideal.
Let H1 and H2 be fractional ideals with denominators d1 and d2 respectively. Let H3 be all finite sums of products of elements from H1 cross H2, and show that H3 is an R module. If H1 and H2 are contained in R then H3 is the product of the two ideals.
Note that the denominator d1d2 drives H3 into R. Thus the product of fractional ideals is another fractional ideal.
Multiplying fractional ideals is both associative and commutative. Also, multiplication by R leaves any fractional ideal unchanged. Therefore the fractional ideals form a commutative monoid.