Modules over algebras vs Modules over Rings?

Question

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.

Answer 1

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.