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The context is lecture notes on Algebra 2 (that is, elementary algebra and not abstract algebra) for a high school student I'm tutoring. Specifically I'm using the term to refer to the field structure of the real numbers - addition, multiplication, associativity, commutativity, distributivity, and polynomials over the real numbers as kind of a catch all for the standard structure on $\mathbb{R}$

To clarify, yes I am aware that an algebra has a very specific definition, and that it does not apply here. However, I have seen this phrase thrown around in the literature, but not with any consistency. If there is a better term you could suggest I would love to hear it too.

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I think your phrasing is absolutely fine, and well-chosen in the sense that what it conveys in (many, at least) people's minds is exactly what you are trying to convey. – Pete L. Clark Jul 15 '11 at 7:52
@crasic: If you think in terms of the traditional use of the word, it is the algebraists' borrowing of the term to refer to a particular kind of structure which is questionable. – André Nicolas Jul 15 '11 at 9:36
I might be inclined to write "algebraic structure" or "algebraic properties". – Nate Eldredge Jul 15 '11 at 12:00
up vote 3 down vote accepted

Both of your rings $\rm\:\mathbb R\:$ and $\rm\:\mathbb R[x]\:$ are in fact instances of an algebra over a field or, more generally, an algebra over ring. Recall that an algebra over a ring $\rm\:R\:$ is simply a ring $\rm\:A\:$ containing a central image of $\rm\:R\:,\:$ i.e. $\rm\:A\:$ contains a subring $\rm\:R' \cong R/I\:$ such that the elements of $\rm\:R'$ commute with all elements of $\rm\:A\:.$

For example $\rm\:R/I\:$ is an $\rm\:R$-algebra for all ideals $\rm\:I\subset R\:,\:$ including $\rm\:R/0\:\cong R\:.\:$ Another well known example are polynomial rings over $\rm\:R\:$ or $\rm\:R/I\:,\:$ where the central image is simply the coefficient subring (the coefficients being central is precisely the condition required to ensure that the evaluation map from polynomial rings is a ring homomorphism and, hence, the fundamental ubiquitous universal property of polynomial rings).

Your examples are also instances of the more general notion of algebraic structures studied in Universal Algebra or General Algebra, i.e. classes of structures with a fixed set of finitary operations satisfying specified (equational) axioms, e.g. monoids, groups, rings, fields.

The latter more general notion of "algebra" encompassess most of the algebraic structures that are studied before abstract algebra, so it is certainly both consistent and acceptable to use the term "algebra" in those elementary contexts.

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Yes, that is fine. $ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, $

If you wish you might use to the arithmetic of the real numbers instead.

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