It is been some time since I studied abstract algebra. But I'm lately teaching some high school students "Algebra". On this level this basically means working with variables, and understanding that certain rules hold for those variables (which represent real numbers). For example:

$$ (a+b)^2 = a^2 +2ab + b^2 \\ a^na^m = a^{n+m} \\ x^2+bx+c = (x+m)(x+n) \quad \text{if } m+n=b \text{ and } mn =c $$

Now in those books that my highschool students read, they never mention for which kind of elements rules like these are true. They just state those rules.

First, I was thinking that this was just sloppy. I would have written something like for real numbers $a,b$ the following rule hold: ...

But then later I realized, wait, those rules are of course much more general. And I can also recall once having learned that there is something like an Algebra, which is some algebraic structure with some rules defined on it.

I maybe see connections that are not really here. But is the subject "Algebra" on highschool called Algebra because the rules they learn are valid for any Algebra as an algebraic structure? Is that also the reason that they don't specify that the rules are valid for real numbers, because the rules are in fact much more general (but they haven't formally learned about those more general structures, so you can not specify it).

  • $\begingroup$ Are you assuming $m$ and $n$ are natural numbers? Then this is a ring. Or integers? Then this is a field. Or rationals? Then I am out of my depth (it would be a field, but a specific type of field I guess). $\endgroup$ – user1729 Feb 27 '18 at 10:02
  • $\begingroup$ I think you can do this for fields. So any set of elements with addition and multiplication defined in a particular way such that it satisfies the properties of a field will have these properties. For example $Z_p$ which is the set of remainders when divided by $p$, where p is a prime, shows these properties. Where addition is defined as $mod((a+b), p)$ and multiplication as $mod((ab), p)$ $\endgroup$ – Piyush Divyanakar Feb 27 '18 at 10:07
  • $\begingroup$ The answer by Matt B is excellent. I just want to add that I've thought about the same problem, and I sometimes do a little extracurricular lesson where I use the metaphor that all our "apps" (integers, real numbers, polynomials, functions: $\Bbb R \rightarrow \Bbb R$) run on "the same operating system", namely "high school algebra" (which in my head is = commutative ring). The formulae you give are "part of the OS" and hold true whatever "app" we load. If I have great students, I mention that there are other operating systems (on which you can run matrices, functions with composition ...). $\endgroup$ – Torsten Schoeneberg Nov 29 '18 at 21:34

I think the structure you want is known as a commutative ring. A ring basically lets you multiply, add and subtract things and end up with something else in the ring again.

Some examples include

  • Real numbers
  • Complex numbers
  • Integers aka whole numbers
  • $n \times n$ matrices (over any ring)
  • Polynomials, i.e. expressions $a_nx^n + a_{n-1}x^{n-1} + \cdots a_0$ (where the $a_i$ live in any ring)

The commutative property means that multiplication doesn't care about order: $a \times b = b \times a$. This is true for the first three examples, but matrices are typically the first case people properly learn where this isn't true. Polynomials will be commutative if the ring where the $a_i$ live is.


Commutativity is important for your top "rule": Note that when you expand the brackets you get $(a+b)^2=(a+b)(a+b)=a^2+ab+ba+b^2$. To get your rule, we must have $ab=ba$ which is precisely the commutativity part. (Oddly enough, you don't actually need commutativity of the polynomial ring for your last "rule" as $ax=xa$ for any $a$, which is why I originally didn't mention it.)

Some others have mentioned fields: these are special commutative rings where you can divide by things as well, so long as it's not zero: the real numbers and complex numbers satisfy this but the integers and polynomials don't.

  • $\begingroup$ Some of the ‘rues’ are valid for commutative rings only. – e.g. $(a+b)^2=\dots$. $\endgroup$ – Bernard Feb 27 '18 at 10:45
  • $\begingroup$ @Bernard: thanks I've edited accordingly. When I thought about commutativity, I only thought about polynomials versus skew-polynomials and decided I didn't need it having missed the top rule! $\endgroup$ – Matt B Feb 27 '18 at 11:56

No, I too think that it's sloppy. For example the "rule" $a^n a^m = a^{n+m}$ does not hold if you consider negative $a$ and non-integer $n$, $m$. "Rules" are not "rules" that one should apply blindly, they are theorems, and theorems have hypotheses. It's crucial to state what these hypotheses are, otherwise it's not mathematics.

Not stating those hypotheses is also setting up the students for failure later, when they encounter settings where the "rules" do not hold (complex numbers for example), or when they encounter more complicated theorems with complicated hypotheses that they need to check (think of the theorems in real analysis). If they don't get used to checking hypotheses, or even knowing that there might be hypotheses to check, they will have a bad time.

As for where those "rules" hold, it depends. The ones you stated hold in a ring, in general (assuming $m$, $n$ are integers). Rings are particular kinds of algebras, namely algebras over the ring $\mathbb{Z}$ (yep, that's a bit circular). But not all "rules" that your students are going to encounter in Algebra come from "rules" in algebras (plural). For example if you ever have a rule involving square roots (e.g. $\sqrt{x^2} = |x|$) then this would be a different setting, because there is no notion of square root in algebras.

Algebras are just one of a plethora of algebraic structure. Algebra (the mathematical field of study) isn't just about algebras (the kind of algebraic structure). You have groups, rings, vector spaces, fields, matrices... Plenty of stuff. Take a look at the Wikipedia page for Algebra. They also have a list of algebraic structure (incomplete, of course).


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