This should be a comment, I guess...
First, a silly remark: every algebra over a field is an algebra over a ring, because fields are rings!
A boolean algebra (in the technical sense of the term, more or less explained in the page http://en.wikipedia.org/wiki/Boolean_algebra_(structure) ) is an algebra over the field of two elemetns $\mathbb F_2$.
On the other hand, Boolean algebra which is the subject of the Wikipedia page you linked to, is not an algebra in that sense. Likewise, Elementary algebra, the subject of the other page you linked to, is also not an algebra in the technical sense.
Boolean algebra and Elementary algebra, as the terms are used in the Wikipedia pages you linked to, are subjects, not algebraic structures.