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.