# What is an algebra?

Is an algebra or 'a algebra' the same thing as an algebraic structure? Or does it have a different meaning?

Thanks

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Do you mean universal algebra? Or algebra over a field? It would really help if you mentioned, where (or in which context) you've encountered the term. – Martin Sleziak Jan 19 '13 at 7:26
BTW I fail to see why this question should be tagged elementary-set-theory. – Martin Sleziak Jan 19 '13 at 7:27
@Martin: Algebra is clearly a set... every question about sets is set theory, no? :-) – Asaf Karagila Jan 19 '13 at 8:40
x is a collection of number, with ring operation – Scott Jan 19 '13 at 11:01

Depends a bit on the context, but "an algebra" is often taken to mean a specific kind of algebraic structure, namely, a vector space with a multiplication operation (or a ring with a vector space structure). See, for example, this Wikipedia entry.

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A couple of years behind but I think it would still be of some pedagogical value.

It is helpful to distinguish between an "Algebra" and an "Algebraic structure", not that such a distinction actually exist but only to make it as clear as possible. ....................................................................................................................................................

Algebra is a very general concept and it is roughly defined as follows:

An algebra A is a set A together with one or more operation o(i) (i=natural numbers). A= {A, O(1), O(2), ....O(n)}

Ex: <A, + , *>

Ex: <{0,1}, |(Sheffer stroke) > this is basically the semantics of first-order logic ....................................................................................................................................................

Algebraic Structures are the familiar structures like groups, fields and rings that are usually taught in a general abstract algebra course. .....................................................................................................................................................

These are basically the same creatures, the difference makes sense as long as we insist on differentiating them based on their immediate significance for the modern mathematics.

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I don't think so. A group is certainly an algebraic structure, but I've never heard anyone refer to a group as an algebra, even though it has an operation. – Gerry Myerson Jun 7 at 22:58