Noetherian rings have the Orzech property #
Main results #
IsNoetherian.injective_of_surjective_of_injective
: ifM
andN
areR
-modules for a ringR
(not necessarily commutative),M
is Noetherian,i : N →ₗ[R] M
is injective,f : N →ₗ[R] M
is surjective, thenf
is also injective.IsNoetherianRing.orzechProperty
: Any Noetherian ring satisfies the Orzech property.
For an endomorphism of a Noetherian module, any sufficiently large iterate has disjoint kernel and range.
Orzech's theorem for Noetherian modules: if R
is a ring (not necessarily commutative),
M
and N
are R
-modules, M
is Noetherian, i : N →ₗ[R] M
is injective,
f : N →ₗ[R] M
is surjective, then f
is also injective. The proof here is adapted from
Djoković's paper Epimorphisms of modules which must be isomorphisms [Djo73],
utilizing LinearMap.iterateMapComap
.
See also Orzech's original paper: Onto endomorphisms are isomorphisms [Orz71].
Orzech's theorem for Noetherian modules: if R
is a ring (not necessarily commutative),
M
is a Noetherian R
-module, N
is a submodule, f : N →ₗ[R] M
is surjective, then f
is also
injective.
Any surjective endomorphism of a Noetherian module is injective.
Any surjective endomorphism of a Noetherian module is bijective.
If M ⊕ N
embeds into M
, for M
noetherian over R
, then N
is trivial.
If M ⊕ N
embeds into M
, for M
noetherian over R
, then N
is trivial.
Equations
Instances For
Any Noetherian ring satisfies Orzech property.
See also IsNoetherian.injective_of_surjective_of_submodule
and
IsNoetherian.injective_of_surjective_of_injective
.