Is there a specific terminology for rings in which we do not require associativity on the multiplication ? Like, for example we say semiring for a ring where we do not require the additive inverses. Any references are appreciated.
Generically, this is normally just called a nonassociative ring.
I don't think there is a common term like "semiring," or at least it is not commonly used.
It is just like "noncommutative ring." There is no cute pithy term, just a straightforward adjective.