Ideal Distributive Law

Distributive Law for Ideals

Given any ideals I,J,K of a ring R, we have that

I(J+K)=IJ+IK.
Proof
I(J+K)=I⟨J∪K⟩=⟨{im:i∈I,m∈J∪K}⟩=⟨{ij:i∈I,j∈J}∪{ik:i∈I,k∈K}⟩=⟨⟨{ij:i∈I,j∈J}⟩∪⟨{ik:i∈I,k∈K}⟩⟩=⟨{ij:i∈I,j∈J}⟩+⟨{ik:i∈I,k∈K}⟩=IJ+IK