An "atom" in Boolean algebra Could someone explain what an atom in Boolean algebra means? I am acquainted with ring theory and group theory but not Boolean algebra. As far as I can tell from browsing around, it is something like a generator, but then again not exactly... Your help will be very much appreciated. Thanks in advance.
Added: It would be particularly helpful if it could be placed in the context of rings or groups. I think there are such entities as Boolean rings...

If $B$ is a Boolean algebra then $x \in B$ is an atom if, for all $y \in B$, either $x \wedge y = x$ or $x \wedge y = 0$. Intuitively, it's a sort of minimal element: the way to think is "if $x$ is an atom and $0 \le y \le x$ then either $y=0$ or $y=x$".
In a Boolean ring $R$, we can think of $\wedge$ as multiplication, so we have that if $x$ is an atom then $xy=x$ or $xy=0$ for all $y \in R$.
For example, let $X$ be a set, let $R = \mathcal{P}X$ be its power set and define set multiplication by intersection and addition by symmetric difference. Then the zero element is $\varnothing$, and the atoms are the elements $A \subseteq X$ such that whenever $Y \subseteq X$ either $A \cap Y = A$ or $A \cap Y = \varnothing$. That is, the atoms of $\mathcal{P}X$ are precisely the singletons $\{ x \}$, and $\varnothing$.
