Leibniz ring

Definition
Depending on convention, the term Leibniz ring may be used for either left-Leibniz ring or right-Leibniz ring.

Left-Leibniz ring
A left-Leibniz ring is a non-associative ring $$R$$ with multiplication denoted by $$[ \, \ ]$$ where the left multiplication by any element is a derivation, i.e., the following holds for all $$a,b,c$$ in the ring:

$$[a,[b,c]] = a,b],c] + [b,[a,c$$

Note that a Lie ring is precisely the same as an alternating ring that is also a left-Leibniz ring.

Right-Leibniz ring
A right-Leibniz ring is a non-associative ring $$R$$ with multiplication denoted by $$[ \, \ ]$$ where the right multiplication by any element is a derivation, i.e., the following holds for all $$a,b,c$$ in the ring:

$$a,b],c] = [[a,c],b] + [a,[b,c$$

Note that a Lie ring is precisely the same as an alternating ring that is also a right-Leibniz ring.

Leibniz algebra
There is an analogous notion of Leibniz algebra which is Leibniz ring that is also a non-associative algebra over some underlying commutative unital ring.