Suppose the stock price $(S_t)$ follows a continuous diffusion which pays dividends with yield $q$. Using different levels of rigour, Kim (1990), Jacka (1991), Myneni (1992) and Carr et al. (1992) derive the following decomposition of the fair values of American options
\begin{align}
C_A(S_0;K,T) &= C_E(S_0;K,T)+ \underbrace{q S_0\int_0^T e^{-q t} \mathbb{Q}_S[\{S_t\geq B_t\}]\text{d}t - rK \int_0^T e^{-rt} \mathbb{Q}[\{S_t\geq B_t\}]\text{d}t}_\text{Early Exercise Premium}, \\
P_A(S_0;K,T) &= P_E(S_0;K,T) + \underbrace{rK \int_0^T e^{-rt} \mathbb{Q}[\{S_t\leq B_t\}]\text{d}t - q S_0\int_0^T e^{-q t} \mathbb{Q}_S[\{S_t\leq B_t\}]\text{d}t}_\text{Early Exercise Premium},
\end{align}
where $B_t$ is the optimal exercise curve and $\mathbb{Q}_S$ is the probability measure which uses $S_te^{qt}$ as numéraire. Mathematically, this decomposition resembles Riesz' decomposition or Doob-Meyer's decomposition.
If the asset pays no dividends, $q=0$, the early exercise premium for a call option would be negative and early exercise is never optimal. A put option, on the other hand, can very well be early exercised if $q=0$.
Pham (1997) and Gukhal (2001) generalise the decomposition to finite-active jump diffusions (you get extra terms capturing the possibility jumping above/below $B_t$).