The Boolean lattice of a Boolean ring

Question

I am proving that a Boolean Ring is also a Boolean Lattice.

I defined $\leq$ as $x\leq y$ when $xy=x$. The supremum is $a+b+ab$, and infimum is $ab$. The Max element is $1$, Min is $0$.

I proved that $\leq$ is an order relation ($x\leq x, (x\leq y \text{ and } y\leq x) \implies x=y, (x\leq y \text{ and } y\leq z) \implies x\leq z$). I proved that $(S, \leq)$ is a lattice, finding supremum and infimum, then I proved that $(S, \leq)$ is bounded (I found min and max).

Now i need to proved that $(S, \leq)$ is distributive, how do I do it? I'm stuck at this last point.

Answer 1

Using $x \vee y = x + y + xy$, and $x \wedge y = xy$,

\begin{align} a \wedge (b \vee c) &= a \cdot (b + c + bc)\\ &= ab + ac + abc\\ &= ab + ac + (ab)(ac)\\ &= (a \wedge b) \vee (a \wedge c). \end{align}