= Kernel
{disambiguate=algebra}
{title2=$\ker f$}
{wiki}
A <kernel (algebra)> records the information lost by a <homomorphism>. For a <group homomorphism> it is the <preimage> of the <identity element>, giving the <kernel of a group homomorphism>; for a <ring homomorphism> or <linear map> it is the preimage of zero, giving the <kernel of a ring homomorphism> or <kernel of a linear map>. More generally, the relation $a\sim b\iff f(a)=f(b)$ records which inputs the homomorphism identifies. In the group case, its <equivalence classes> are <cosets> of the <group kernel>, as explained by <fibres of a group homomorphism are cosets of its kernel>.
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