Editing Kernel (algebra) (section)

=== Ring homomorphisms ===

Consider the mapping <math> \varphi : \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} </math> where the later ring is the integers modulo 2 and the map sends each number to its [[Parity (mathematics)|parity]]; 0 for even numbers, and 1 for odd numbers. This mapping turns out to be a homomorphism, and since the additive identity of the later ring is 0, the kernel is precisely the even numbers.<ref name="Dummit Ring Kernel Examples">{{harvnb|Dummit|Foote|2004|p=240}}</ref>

Let <math> \varphi: \mathbb{Q}[x] \to \mathbb{Q} </math> be defined as <math>\varphi(p(x))=p(0)</math>. This mapping , which happens to be a homomorphism, sends each polynomial to its constant term. It maps a polynomial to zero [[if and only if]] said polynomial's constant term is 0.<ref name="Dummit Ring Kernel Examples" /> Polynomials with real coefficients can receive a similar homomorphism, with its kernel being the polynomials with constant term 0.<ref>{{harvnb|Hungerford|2014|p=155}}</ref>