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!