Understanding the difference between a module and an algebra. So I am try to really understand the difference between the two, and looking things up online, I came across this https://www.physicsforums.com/threads/difference-between-algebra-and-module.149042/#:~:text=%20algebra%3A%20vector%20space%20with%20the%20multiplication%20which,vector%20space%20with%20a%20multiplication%20of%20vectors%20defined. which claims that..
The set of all polynomials over the real numbers is an algebra that is not a module. The set of all pairs $(x,y)$ with addition defined by $(x,y)+ (u,v) = (x+u,y+v), x, y, u, v$ all members of $\mathbb{Z}_4$ (integers mod $4$) is a module that is not an algebra.
I am confused, because isn't the set of all polynomials over the real numbers an abelian group, and therefore would have to be a module? And then isn't $\phi : \mathbb{Z} \rightarrow \mathbb{Z}_4 \times \mathbb{Z}_4$ such that $\phi(z) = (z/4\mathbb{Z},0)$ a homomorphism from $\mathbb{Z}$ to the center of $\mathbb{Z}_4 \times \mathbb{Z}_4$, and thus $\mathbb{Z}_4 \times \mathbb{Z}_4$ is an algebra?
Source https://www.physicsforums.com/threads/difference-between-algebra-and-module.149042/
 A: I think it's confusing to call something "a module" or "an algebra". I find it much clearer to say "a module over $R$" or "an algebra over $F$". In particular, the set $\mathbb{R}[X]$ of real polynomials can be viewed as a vector space over $\mathbb{R}$ (the field), and hence as a module over $\mathbb{R}$ (the ring). With polynomial multiplication, it is an $\mathbb{R}$-algebra too.
The user you reference is either very confused or very pedantic. If $F$ is a field, every $F$-algebra is in particular an $F$-vector space and hence an $F$-module. The only possible reason to object here is if we distinguish between $F$ as a field and $F$ as a ring. In theory there could be a distinction - if you define a field as a tuple $(X, +,  \times, -, {}^{-1}, 0, 1)$ where $-: a \to -a$, and  ${}^{-1}: a \to a^{-1}$, then a ring and a field are different (because rings don't come with multiplicative inversion).
But that's a very pedantic view, if it's even correct. In reality, most people agree that "every field is a ring", and not "every field is isomorphic to a ring" (or something like this). It's also very sensitive to definitions: you could define fields in such a way that they literally are rings. But, in summary: Every algebra is a vector space, and therefore a module.
A module, given to you by itself, is not an algebra. For one thing, the module must be a vector space, conventionally. Moreover, it may be possible to make it into an algebra, by giving it a multiplication - but there might be many different multiplications, making that vector space into many different algebras. So you have to say what multiplication you're thinking of, before you can verify whether it's an algebra or not.
A: Some of the information in the sources you quote is misleading. Basically an algebra over a ring is a module with the extra structure of a multiplication satisfying various axioms. So the set of all polynomials over the real numbers can certainly be regarded as a module: just forget about the multiplication. Also the set of all pairs from $\mathbb{Z}_4$ can certainly be regarded as an algebra over $\mathbb{Z}$  or over $\mathbb{Z}_4$ if you define the multiplication suitably.
