When to keep last month's rent?

[Megan R.]

Hi everyone, I could use some help because i dont know anything about  about rentals and my mom is having some issues.  My mother has her first tenant and the one year lease expired today (sept 1st).  Her tenant asked her on August 1st if another year lease could be signed and paid her the rent for august.  On August 14th the tenant informed her that she would be moving on the 31st and did not want to renew anymore.  A friend of my moms with rental experience has told her that she can keep the last months rent that was paid upon lease signing because the tenant didn't give her 30 days notice.  I have no idea if this is accurate and i dont want my mother getting into any trouble or being taken advantage of.  Any help would be greatly appreciated!