distributivity

Given a set $S$ with two binary operations $+\colon S \times S \to S$ and $\cdot\colon S \times S \to S$ , we say that $\cdot$ is right distributive over $+$ if $$(a+b) \cdot c = (a \cdot c) + (b \cdot c)\mathrm{~for~all~} a,b,c \in S$$ and left distributive over $+$ if $$a \cdot (b+c) = (a \cdot b) + (a \cdot c)\mathrm{~for~all~}a,b,c \in S.$$ If $\cdot$ is both left and right distributive over $+$ , then it is said to be distributive over $+$ (or, alternatively, we may say that $\cdot$ distributes over $+$ ).