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An eta-quotient of level $N$ is a modular form of the shape $f(z) =\prod_{\delta | N} \eta(\delta z)^{r_{\delta}}$. We study the problem ofdetermining levels $N$ for which the graded ring of holomorphic modular formsfor $\Gamma_{0}(N)$ is generated by (holomorphic, respectively weaklyholomorphic) eta-quotients of level $N$. In addition, we prove that if $f(z)$is a holomorphic modular form that is non-vanishing on the upper half plane andhas integer Fourier coefficients at infinity, then $f(z)$ is an integermultiple of an eta-quotient. Finally, we use our results to determine thestructure of the cuspidal subgroup of $J_{0}(2^{k})(\mathbb{Q})$.

On spaces of modular forms spanned by eta-quotients