Kernel (algebra)
Invarious branchesmathematics that fall underheadingabstract algebra,kernel ofhomomorphism measuresdegreewhichhomomorphism failsbe injective.
The definitionkernel takes various formsvarious contexts. Butallthem,kernel ofhomomorphismtrivial (insense relevantthat context) ifonly ifhomomorphisminjective. The fundamental theorem on homomorphisms (or first isomorphism theorem) istheorem, again taking various forms, that applies toquotient algebra defined bykernel.
In this article, we first survey kernelssome important typesalgebraic structures; then we give general definitions from universal algebrageneric algebraic structures.
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2 Universal algebra 3 Algebrasnonalgebraic structure 4 Kernelscategory theory |
Surveyexamples
Linear operators
Let VW be vector spaceslet T belinear transformation from VW. If 0W iszero vectorW, thenkernelT ispreimage ofsingleton set {0W}; that is,subsetV consistingall those elementsV thatmapped by T toelement 0W. The kernelusually denoted "ker T" (orvariation). In symbols:
It turns out that ker TalwayssubspaceV. Thus,makes sensespeak ofquotient space V/(ker T). The first isomorphism theoremvector spaces states that this quotient spacenaturally isomorphic toimageT (which issubspaceW). Asconsequence,dimensionV equalsdimension ofkernel plusdimension ofimage.
If VWfinite-dimensionalbases have been chosen, then T can be described bymatrix M, andkernel can be computed by solvinghomogenous systemlinear equations Mv = 0. In this representation,kernel corresponds tonullspaceM. The dimension ofnullspace, callednullityM,given bynumbercolumnsM minusrankM, asconsequence ofrank-nullity theorem.
Solving homogeneous differential equations often amountscomputingkernelcertain differential operators. For instance,orderfind all twice-differentiable functions f fromreal lineitself such that
- xf''(x) + 3f'(x) = f(x),
- (Af)(x) = xf''(x) + 3f'(x) - f(x)
One can define kernelshomomorphisms between moduless overringan analogous manner. This includes kernelshomomorphisms between abelian groups asspecial case. This example capturesessencekernelsgeneral abelian categories; see Kernel (category theory).
Group homomorphisms
Let GH be groupsslet f begroup homomorphism from GH. If eH isidentity elementH, thenkernelf ispreimage ofsingleton set {eH}; that is,subsetG consistingall those elementsG thatmapped by f toelement eH. The kernelusually denoted "ker f" (orvariation). In symbols:
It turns out that ker fnot onlysubgroupG butfactnormal subgroup. Thus,makes sensespeak ofquotient group G/(ker f). The first isomorphism theoremgroups states that this quotient groupnaturally isomorphic toimagef (which issubgroupH).
Inspecial caseabelian groups, this worksexactlysame way as inprevious section.
Ring homomorphisms
Let RS be ringsslet f bering homomorphism from RS. If 0S iszero elementS, thenkernelf ispreimage ofsingleton set {0S}; that is,subsetR consistingall those elementsR thatmapped by f toelement 0S. The kernelusually denoted "ker f" (orvariation). In symbols:
It turns out that, although ker fgenerally notsubringR, itneverthelesstwo-sided idealR. Thus,makes sensespeak ofquotient ring R/(ker f). The first isomorphism theoremrings states that this quotient ringnaturally isomorphic toimagef (which issubringS).
To some extent, this can be thoughtasspecial case ofsituationmodules, since theseall bimodules overring R:
- R itself;
- any two-sided idealR (such as ker f);
- any quotient ringR (such as R/(ker f); and
- codomainany ring homomorphism whose domainR (such as S,codomainf).
This example capturesessencekernelsgeneral Mal'cev algebras.
Monoid homomorphisms
Let MN be monoidsslet f bemonoid homomorphism from MN. Thenkernelf issubset ofdirect product M × M consistingall those ordered pairselementsM whose componentsboth mapped by f tosame elementN. The kernelusually denoted "ker f" (orvariation). In symbols:
It turns out that ker fan equivalence relation on M,in factcongruence relation. Thus,makes sensespeak ofquotient monoid M/(ker f). The first isomorphism theoremmonoids states that this quotient monoidnaturally isomorphic toimagef (which issubmonoidN).
Thisvery differentflavour fromabove examples. In particular,preimage ofidentity elementNnot enoughdeterminekernelf. Thisbecause monoidsnot Mal'cev algebras.
Universal algebra
Allabove cases may be unifiedgeneralizeduniversal algebra.
General case
Let AB be algebraic structures ofgiven typelet f behomomorphismthat type from AB. Thenkernelf issubset ofdirect product A × A consistingall those ordered pairselementsA whose componentsboth mapped by f tosame elementB. The kernelusually denoted "ker f" (orvariation). In symbols:
It turns out that ker fan equivalence relation on A,in factcongruence relation. Thus,makes sensespeak ofquotient algebra A/(ker f). The first isomorphism theoremgeneral universal algebra states that this quotient algebranaturally isomorphic toimagef (which issubalgebraB).
Note thatdefinitionkernel here (as inmonoid example) doesn't depend onalgebraic structure; it'spurely set-theoretic concept. For more on this general concept, outsideabstract algebra, see Kernel (function).
Mal'cev algebras
IncaseMal'cev algebras, this construction can be simplified. Every Mal'cev algebra hasspecial neutral element (the zero vector incasevector spaces,identity element incasegroupss, andzero element incaseringss or moduless). The characteristic feature ofMal'cev algebrathat we can recoverentire equivalence relation ker f fromequivalence class ofneutral element.
To be specific, let AB be Mal'cev algebraic structures ofgiven typelet f behomomorphismthat type from AB. If eB isneutral elementB, thenkernelf ispreimage ofsingleton set {eB}; that is,subsetA consistingall those elementsA thatmapped by f toelement eB. The kernelusually denoted "ker f" (orvariation). In symbols:
The notionideal generalisesany Mal'cev algebra (as subspace incasevector spaces, normal subgroup incasegroups, two-sided ring ideal incaserings,submodule incasemoduless). It turns out that although ker f may not besubalgebraA, itnevertheless an ideal. Thenmakes sensespeak ofquotient algebra G/(ker f). The first isomorphism theoremMal'cev algebras states that this quotient algebranaturally isomorphic toimagef (which issubalgebraB).
The connection between this andcongruence relation ismore general typesalgebrasas follows. First,kernel-as-an-ideal isequivalence class ofneutral element eA underkernel-as-a-congruence. Forconverse direction, we neednotionquotient inMal'cev algebra (whichdivision on either sidegroupssubtractionvector spaces, modules,rings). Using this, elements aa'Aequivalent underkernel-as-a-congruence ifonly if their quotient a/a'an element ofkernel-as-an-ideal.
Abelian algebras
I needlook up more stuff on universal algebra.
Algebrasnonalgebraic structure
Sometimes algebrasequipped withnonalgebraic structureadditiontheir algebraic operations. For example, one may consider topological groups or topological vector spaces, withequipped withtopology. In this case, we would expecthomomorphism fpreserve this additional structure; intopological examples, we would want fbecontinuous map. The process may run intosnag withquotient algebras, which may not be well-behaved. Intopological examples, we can avoid problems by requiring that topological algebaic structures be Hausdorff (asusually done); thenkernel (however itconstructed) will beclosed set andquotient space will work fine (and also be Hausdorff).
Kernelscategory theory
The notionkernelcategory theory isgeneralisation ofkernelsabelian algebras; see Kernel (category theory). The categorical generalisation ofkernel ascongruence relation iskernel pair. (Therealsonotiondifference kernel, or binary equaliser.)
