More operations on modules and ideals
N.annihilator is the ideal of all elements
r : R such that
r • N = 0.
N.colon P is the ideal of all elements
r : R such that
r • P ⊆ N.
The homomorphism from
R/(⋂ i, f i) to
∏ i, (R / f i) featured in the Chinese
Remainder Theorem. It is bijective if the ideals
f i are comaximal.
Chinese Remainder Theorem. Eisenbud Ex.2.6. Similar to Atiyah-Macdonald 1.10 and Stacks 00DT
The radical of an ideal
I consists of the elements
r such that
r^n ∈ I for some
Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6.
I.map f is the span of the image of the ideal
f, which may be bigger than
the image itself.
I.comap f is the preimage of
comap are adjoint, and the composition
map f ∘ comap f is the
The map on ideals induced by a surjective map preserves inclusion.
Special case of the correspondence theorem for isomorphic rings
A proper ideal
I is primary iff
xy ∈ I implies
x ∈ I or
y ∈ radical I.
Kernel of a ring homomorphism as an ideal of the domain.
An element is in the kernel if and only if it maps to zero.
If the target is not the zero ring, then one is not in the kernel.
The kernel of a homomorphism to an integral domain is a prime ideal.
The ring hom
R/J →+* S/I induced by a ring hom
f : R →+* S with
J ≤ f⁻¹(I)
h are kept as seperate hypothesis since H is used in constructing the quotient map
If we take
J = I.comap f then
quotient_map is injective automatically.
Commutativity of a square is preserved when taking quotients by an ideal.
lift_of_surjective f hf g hg is the unique ring homomorphism
- such that
φ.comp f = g (
f : A →+* B is surjective (
g : B →+* C satisfies
hg : f.ker ≤ g.ker.
lift_of_surjective_eq for the uniqueness lemma.
f | \ g
B ----> C