It's hard to tell from the pic what the piece on the left might be -- possibly bone, possibly Bakelite, etc. Perhaps a well lit close-up, front and back, will show more detail.
And the pre-Columbian style brooch appears to be dyed onyx, sometimes referred to as 'Mexican Jade', however, it is not a true jade. The setting on this place is likely silver, which often has a high fineness as typical of the period (circa 1940s).