Difference between revisions of "Kernel"

Revision as of 12:46, 16 June 2008

In general, a kernel is a measure of the failure of a homomorphism to be injective.

In set theory, if $S$ and $T$ are sets, with $f$ a function mapping $S$ into $T$ , the kernel of $f$ is quotient set of $S$ under the equivalence relation $R(x,y)$ defined as " $f(x)=f(y)$ ".

In algebra, a kernel is generally the inverse image of an identity element under a homomrphism. For instance, in group theory, if $G$ and $H$ are groups, and $f : G \to H$ is a homomorphism of groups, the kernel of $f$ is the set of elements of $G$ that map to the identity of $H$ , i.e., the set $f^{-1}(e_{H})$ . The kernel is a normal subgroup of $G$ , and in fact, every normal subgroup of $G$ 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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 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 [[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.
+In [[abstract algebra |algebra]], a kernel is generally the inverse image of an identity element under a [[homomrphism]].  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.
 {{stub}}
 [[Category:Abstract algebra]]

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Difference between revisions of "Kernel"

Revision as of 12:46, 16 June 2008