The main effect will be non-significant if the interaction is significant? I am using linear mixed models to identify important factors, and it turns out that:


*

*A: significant 

*B: not significant

*A×B: significant


Does it mean that because A×B shows that the effect of A depends on the effect of B, only the effect of A is not actually significant?
I have read many sources, and they seem to suggest that if the effect of A×B is significant, then we cannot interpret that the effect of A is significant on our dependent variable. Am I understanding right?
 A: What you read is correct. If the interaction is significant, interpreting either main effect, whether significant or not, is basically pointless (and misleading). The reason is that when $A$ and $B$ are involved in an interaction, the coefficient for $A$ is the effect of $A$ when $B=0$; in other words, the effect is conditional on the value of $B$, and is not a main effect. Similarly, the coefficient for $B$ is the effect of $B$ when $A=0$.
The fact that $A$ is significant merely means that $A$ has an effect when $B=0$. Similarly, the fact that $B$ is not significant merely means that $B$ doesn't have an effect when $A=0$, though it probably does have an effect for other values of $A$; this is precisely why the interaction is significant.
What you would need to do is look at simple slopes, which shows the significance of the $A$ effect as a function of the $B$ variable. You need to determine at which values of $B$ does $A$ have an effect, and vice-versa. Kris Preacher provides an online tool to decompose 2-way interactions in linear mixed models.
