I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse.
Is there something similar except with no requirement for an additive inverse. For example, all the non-negative rational numbers. Every number other than 0 has a multiplicative inverse, but no additive inverses. We have both the multiplicative and additive identities. Multiplication is still associative over addition.
Is there a name for such an algebraic structure, and has it been studied the way rings have?
The relevant concept here is semiring, or semifield if you include multiplicative inverses.