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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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    You need a group structure for the addition.2012-06-05
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    $A,B$ are matrices?2012-06-05
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    @PrasadG: If they were matrices, YKY would have told us that it was left-distributive as well as right-distributive, so presumably not.2012-06-05
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    Are there additive or multiplicative identities? If there are, then you *almost* have a semiring. In any case, you should not call it a ring unless it is one. **Edit**: Apparently there's something called a "near-ring"; this is closer to that than a semiring. Perhaps you should check if it might be a near-ring. Or maybe just a near-semiring, since that's apparently a known term as well. That would require very little extra.2012-06-05
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    To answer your first question: do not call it a ring.2012-06-05
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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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