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According to the definitions of algebra over a ring and algebra over a field, are Elementary Algebra and Boolean Algebra examples of algebra over a ring or algebra over a field? How can one show that? What are the ring or field that the algebras are with respect to?

Thanks and regards!

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2 Answers 2

up vote 4 down vote accepted

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.

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Totally agree. There is a not so obvious difference between algebra and algebra. –  AD. Sep 6 '10 at 14:46
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Disclaimer: I still don't have a great grasp on a lot of abstract algebra, but I'd like to try and help when I can.

EDIT: I think Mariano Suárez-Alvarez's comment was more to the point by addressing the issues of structures vs. subjects

The trouble with the boolean algebra you linked to is that there aren't inverse operations and neutral elements to union and intersection because with intersection and union each element is its own neutral element while the axioms of a field or ring state that there is one neutral element that works for your whole ring or field. Without neutral elements, you can't define inverses because the result of 'multiplying' or 'adding' an element with its inverse element is supposed to be the multiplicatively neutral or additively neutral element. So boolean algebra can't be an algebra over a ring or an algebra over a field since it is neither a ring nor field since there are no neutral elements.

On the other hand elementary algebra (assuming you are working with either the real or complex numbers) is a ring and also a field. We can make elementary algebra into an algebra over a field by letting the vector space be the field itself and the axioms will be satisfied; however, this is technically adding additional structure to elementary algebra (the vector space) and so its probably no technically considered 'elementary algebra.'

Please let me know if there are any unclear points or errors :)

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