Difference between revisions of "Kernel"
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In [[set theory]], if <math>S</math> and <math>T</math> are [[set]]s, with <math>f</math> a [[function]] mapping <math>S</math> into <math>T</math>, the '''kernel''' of <math>f</math> is [[quotient set]] of <math>S</math> under the equivalence relation <math>R(x,y)</math> defined as "<math>f(x)=f(y)</math>". | In [[set theory]], if <math>S</math> and <math>T</math> are [[set]]s, with <math>f</math> a [[function]] mapping <math>S</math> into <math>T</math>, the '''kernel''' of <math>f</math> is [[quotient set]] of <math>S</math> under the equivalence relation <math>R(x,y)</math> defined as "<math>f(x)=f(y)</math>". | ||
| − | In [[abstract algebra |algebra]], a kernel is generally the inverse image of an identity element. For instance, in [[group theory]], if <math>G</math> and <math>H</math> are [[group]]s, and <math>f : G \to H</math> is a | + | In [[abstract algebra |algebra]], a kernel is generally the inverse image of an identity element under a [[homomorphism]]. For instance, in [[group theory]], if <math>G</math> and <math>H</math> are [[group]]s, and <math>f : G \to H</math> is a homomorphism of groups, the kernel of <math>f</math> is the set of elements of <math>G</math> that map to the [[identity]] of <math>H</math>, i.e., the set <math>f^{-1}(e_{H})</math>. The kernel is a [[normal subgroup]] of <math>G</math>, and in fact, every normal subgroup of <math>G</math> is the kernel of a homomorphism. Similarly, in [[ring theory]], the kernel of a homomorphism is the inverse image of zero; the kernel is a two-sided [[ideal]] of the ring, and every two-sided ideal of a ring is the kernel of a ring homomorphism. |
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[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
Latest revision as of 12:47, 16 June 2008
In general, a kernel is a measure of the failure of a homomorphism to be injective.
In set theory, if 
and 
are sets, with 
a function mapping into , the kernel of is quotient set of under the equivalence relation 
defined as "
".
In algebra, a kernel is generally the inverse image of an identity element under a homomorphism. For instance, in group theory, if 
and 
are groups, and 
is a homomorphism of groups, the kernel of is the set of elements of that map to the identity of , i.e., the set 
. The kernel is a normal subgroup of , and in fact, every normal subgroup of is the kernel of a homomorphism. Similarly, in ring theory, the kernel of a homomorphism is the inverse image of zero; the kernel is a two-sided ideal of the ring, and every two-sided ideal of a ring is the kernel of a ring homomorphism.
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