I have read that subgroups, subrings, submodules, etc. are substructures.
But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals and Noetherian modules are defined with submodules. Isn't it awkward? Why does submodule correspond to ideal, not subring? Is there any definition of Noetherian with subrings?
As I'm studying commutative algebra, it looks like ideals are more important than subrings. But why is it ideal, not subring (which seems to correspond to all other substructures)? Though I am not very familiar with pseudo-rings, is it true that ideal is a sub-pseudo-ring (or sub-rng) and thus we can view ideal as a kind of substructure?
Best Answer
The "right" notion of a substructure of an algebraic gadget is the kernel of a homomorphism. For abelian groups, and more generally modules, these are subgroups, respectively submodules. For groups, we need normal subgroups. For rings, we need ideals.