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?

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    $\begingroup$ The word you're looking for is semiring (or if you want multiplicative inverses, then semifield) $\endgroup$ – zcn Jan 20 '14 at 1:21
  • $\begingroup$ Thanks. Semifield is exactly what I was looking for. How do I accept your answer? I don't see a checkmark anywhere. I'm a new member, does that mean I can't accept answers immediately? $\endgroup$ – Ameet Sharma Jan 20 '14 at 1:34
  • $\begingroup$ You can't accept comments as answers: you can always accept answers to your own questions though. If you like, I can post this as an answer $\endgroup$ – zcn Jan 20 '14 at 1:48
  • $\begingroup$ Ok. Thanks. sounds good. $\endgroup$ – Ameet Sharma Jan 20 '14 at 1:50

The relevant concept here is semiring, or semifield if you include multiplicative inverses.


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