Since I've started module theory I was confused with a point. What is more general, the theory of modules over algebras or over arbitrary rings?

I hope this is not a pointless question so let me try to explain:

  • Every ring $R$ is a $\mathbb Z$-algebra so when we study modules over algebras we are also studing over arbitrary rings;
  • On the other hand, if $A$ is a $R$-algebra then $A$ is also a ring so when we study modules over arbitrary rings we are also considering this case. Of course, in this case $A$ has more properties which can be be taken into account.

I hope I made clear my point.

Obs. When I say $A$ is a $R$-algebra it must be understood $A$ is a ring which is also a $R$-module where $R$ is a commutative right with identify (not necessarily a field) such that the ring and module structures are compatible in the obvious way.

Thanks.

  • To your first point, while every ring is "a $\mathbb{Z}$-algebra", the term "algebra" is often used to mean an algebra over a field, giving a more specialized meaning. One might also observe that a ring $R$ is an $R$-algebra, giving no additional structure. If by algebra you mean something more specialized than an arbitrary ring, then modules over algebras would be a narrower topic than modules over rings. – hardmath Jun 26 '16 at 13:47
  • Sorry, I should have been more precise as to my definition of algebras. I'll fix it. – PtF Jun 26 '16 at 15:59
  • 2
    @hardmath only commutative rings can be understood as algebras over themselfs – Jakob Werner Jun 26 '16 at 16:05

What is more general, the theory of modules over algebras or over arbitrary rings?

Yes, every ring is a $\Bbb Z$-algebra, and its modules are automatically $\Bbb Z$-algebra modules. And of course every $\Bbb Z$-algebra is a priori a ring. With this in mind, then neither theory is more general than the other: they are the same.

The class of algebras which are most distinguished from others are the finite dimensional algebras over fields. Why? They can be represented as subrings of square matrix rings, and thus there are many techniques to analyze them. That's one of the things that makes the theory of algebras over fields more special than the general theory of $\Bbb Z$-algebras.

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