Sufficient condition for martingale property

Question

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ \mathbb{E}[M_s|\mathcal{F}_t]=M_t $$ $M$ is said to be a martingale. Now let assume I can integrate with respect to $M$ (say $M$ is a semi-martingale and I am using Itô integration). Then $$ \forall t < s, \ \mathbb{E}[\int_t^s dM_u | F_t] = 0 $$ still implies that $M$ is a martingale. What about $$ \forall t < s, \ \mathbb{E}[\int_t^s \alpha_u dM_u | F_t] = 0 $$ where $\alpha$ is a predictable stochastic process ? Under which conditions on $\alpha$ (except being constant) is $M$ still a martingale ?