Kernel (algebra)

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, the kernel of a <a href="/facts/Homomorphism/0gyqdEse">homomorphism</a> is the relation describing how elements in the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of the homomorphism become related in the <a href="/facts/Image_(mathematics)/KANefMAl">image</a>. A homomorphism is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that preserves the underlying <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> in the domain to its image.
When the algebraic structures involved have an underlying <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> structure, the kernel is taken to be the <a href="/facts/Preimage/KANefMAl">preimage</a> of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity. For example, the map that sends every <a href="/facts/Integer/PUwwrawx">integer</a> to its <a href="/facts/Parity_(mathematics)/7zq2UM4V">parity</a> (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers <a href="/facts/Modulo/5tKQmM76">modulo</a> 2, and its respective kernel would be the even integers which all have 0 as its parity. The kernel of a homomorphism of group-like structures will only contain the identity if and only if the homomorphism is <a href="/facts/Injective_function/5ES6tqf5">injective</a>, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.
For some types of structure, such as <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a> and <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>, the possible kernels are exactly the substructures of the same type. This is not always the case, and some kernels have received a special name, such as <a href="/facts/Normal_subgroups/aRDaGwLm">normal subgroups</a> for groups and <a href="/facts/Two-sided_ideal/vCvytduX">two-sided ideals</a> for <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>. The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a <a href="/facts/Congruence_relation/TV3njpqF">congruence relation</a>.
Kernels allow defining <a href="/facts/Quotient_object/lEcCIhrP">quotient objects</a> (also called <a href="/facts/Quotient_(universal_algebra)/8uQzYkm7">quotient algebras</a> in <a href="/facts/Universal_algebra/uMJAL6ER">universal algebra</a>). For many types of algebraic structure, the <a href="/facts/Fundamental_theorem_on_homomorphisms/1frDbpLj">fundamental theorem on homomorphisms</a> (or <a href="/facts/First_isomorphism_theorem/SNpPTkrd">first isomorphism theorem</a>) states that <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of a homomorphism is <a href="/facts/Isomorphism/pj8KgkxU">isomorphic</a> to the quotient by the kernel.

Kernel (algebra) open-in-new

Kernel (algebra)