It involves a serial killer who kills women and scalps them and takes the ring finger so that he can put his mothers ring on. The serial killer was also abused by his mother and through out the movie they play a song over and over again or every time he kills another woman.

I couldn't find any references to taking their ring fingers but maybe Maniac (1980)

Its not that movie , when the killer would kill people a some what sad song would be played