Why is the coefficient of a fraction $\frac{1}{denominator}$ instead of simply 1?

Question

Why is the coefficient of a fraction $\frac{1}{denominator}$ instead of simply $1$?

Wouldn't the result be the same?

Answer 1

Depends of where you are taking the coefficients. For example Q is generated by 1, taking the coefficients in Q, but if you take them in Z you get just the integers and not all rationals. This becomes clearer when one studies modules and vector spaces.