Distributive property

Remarks

Example: matrix multiplication on the set of $n \times n$ real matrices is distributive over matrix addition, since $A(B + C) = AB + AC$ and $(B + C)A = BA + CA$ by the entrywise formulas.

Non-example: on the set $\R_{>0}$ of positive reals, take ordinary multiplication $\cdot$ in place of $+$, and let $a \ast b = a^b$. Then $\ast$ is right distributive over $\cdot$, since $(bc)^a = b^a c^a$. It is not left distributive, since $2^{1 \cdot 1} = 2$ while $2^1 \cdot 2^1 = 4$.

If $\ast$ is commutative, left and right distributivity are equivalent. In a ring, multiplication is distributive over addition; in a Boolean algebra, each of $\wedge$ and $\vee$ is distributive over the other.