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If I have an algebraic structure obeying these rules:

  • non-commutative multiplication: $ A*B \neq B*A $.
  • commutative addition: $ A+B = B+A $.
  • associative addition and multiplication: $ (A+B)+C = A+(B+C) \quad \mbox{ and }\quad (A*B)*C = A*(B*C) .$
  • distribution on the right: $ (A+B)*C = A*C+B*C $.

The elements need not be numbers (I'm using this structure in my A.I. research).

Is it OK if I call it a non-commutative ring? Or how should I call such a structure?

Thanks!

EDIT: I think $0$ and $1$ can be added to it, though I don't see their significance in my application yet. Also I realize that in my structure + is idempotent: $ A+A = A $.

Adding left distribution does not seem to affect my application, so I guess I can call it a semi-ring. Thanks for the answers!

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    @Prasad: Not necessarily matrices, but that is one potential interpretation that can provide powerful techniques. I'm actually representing natural language sentences using this structure and the elements are words or "concepts".2012-06-07

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If the additive structure is a group, i.e. additive inverses exist, then you have a near-ring. If not, but you have both distributive laws, and addition is commutative, then you have a semiring. If neither, then it's generalization of one of these structures (possibly without a standard name).

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    @Harry Iiirc, there used to be other name(s) for that, which is why I said possibly without a standard name. In any case, the point was to give the OP the terminology need to do searches (your comment did not exist when I started writing the answer).2012-06-05