# Is this algebra a non-commutative ring?

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. – Davide Giraudo Jun 5 '12 at 13:51
$A,B$ are matrices? – Prasad G Jun 5 '12 at 13:52
@PrasadG: If they were matrices, YKY would have told us that it was left-distributive as well as right-distributive, so presumably not. – MJD Jun 5 '12 at 13:53
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. – Harry Altman Jun 5 '12 at 13:56
To answer your first question: do not call it a ring. – rschwieb Jun 5 '12 at 16:50

## 1 Answer

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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Well, there's near-semiring, though that would still require an additive identity and 0a=0: en.wikipedia.org/wiki/Near-semiring – Harry Altman Jun 5 '12 at 14:03
@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). – Bill Dubuque Jun 5 '12 at 14:07