Glossaryring theory
Ring theory isbranchmathematicswhich ringsstudied: that is, structures supporting both an addition andmultiplication operation. This isglossarysome terms ofsubject.
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2 Typeselements 3 Homomorphismsideals 4 Typesrings 5 Miscellaneous |
Definition ofring
A ringan abelian group (R,+) togetheran associative operation * whichdistributive over +has an identity element 1respect*. The operation +referred asaddition*referred asmultiplication. The identity elementrespect+written as 0.
The ringjust one elementcalledtrivial ring.
;Characteristic : The characteristic ofring issmallest positive integer n satisfying n1=0 ifexists0 otherwise. In particular ne=0all elements e ofring.
Typeselements
; Idempotent : An element e ofringidempotent if e2 = e.
; Central : An element r ofring Rcentral if xr = rxall xR. The setall central elements formssubringR, known ascenterR.
; Nilpotent : An element rRnilpotent if there existspositive integer n such that rn = 0.
; Unit or invertible element : An element r ofring R isunit if there exists an element r-1 such that rr-1=r-1r=1. This element r-1uniquely determined by ris calledmultiplicative inverser. The setunits formsgroup under multiplication.
; Zero divisor : A nonzero element rRsaidbezero divisor if there exists s ≠ 0 such that sr=0 or rs=0. Ifring hasZero divisor whichalsounit, thenring has no other elementsistrivial ring.
Homomorphismsideals
; Factor ring : Givenring Ran ideal IR,factor ring isset R/Icosets {a+I : a∈R} togetheroperations (a+I)+(b+I)=(a+b)+I(a+I)*(b+I)=ab+I. The relationship between ideals, homomorphisms,factor ringssummed up infundamental theorem on homomorphisms.
; Finitely generated ideal : A left ideal Ifinitely generated if there exist finitely many elements a1,...,an such that I = Ra1 + ... + Ran. A right ideal Ifinitely generated if there exist finitely many elements a1,...,an such that I = a1R + ... + anR. A two-sided ideal Ifinitely generated if there exist finitely many elements a1,...,an such that I = Ra1R + ... + RanR.
; Ideal : A left ideal IR issubgroup or (R,+) such that aI ⊆ Iall a∈R. A right ideal issubgroup(R,+) such that Ia⊆Iall a∈R. An ideal (sometimesemphasis:two-sided ideal) issubgroup whichbothleft ideal andright ideal.
; Jacobson radical : The intersectionall maximal left ideals inring formstwo-sided ideal,Jacobson radical ofring.
; Kernel ofring homomorphism : It ispreimage0 incodomain ofring homomorphism. Every ideal iskernel ofring homomorphismvice versa.
; Maximal ideal : A left ideal ofring R whichnot containedany other left ideal but R itselfcalledmaximal left ideal. Maximal right idealsdefined similarly. In commutative rings, thereno difference,one speaks simplymaximal ideals.
; Nilradical : The setall nilpotent elements incommutative ring forms an ideal,nilradical ofring. The nilradicalequal tointersectionallmaximal ideals,also equal tointersectionallprime ideals.
; Prime ideal : An ideal P incommutative ring Rprime if P ≠ Rifall abRabP, we have aP or bP. Every maximal ideal incommutative ringprime.
; Principal ideal :principal left ideal inring R isleft ideal ofform Rasome element aR;principal right ideal isright ideal ofform aRsome element aR;principal ideal istwo-sided ideal ofform RaRsome element aR''.
; Radicalan ideal : The radicalan ideal I incommutative ring consistsall those ring elementspowerwhich liesI. Itequal tointersectionall maximal ideals containing I.
; Ring homomorphism : A function f : R → S between rings (R,+,*)(S,⊕,×) isring homomorphism ifhasspecial properties that
- f(a + b) = f(a) ⊕ f(b)
- f(a * b) = f(a) × f(b)
- f(1) = 1
- for any elements abR.
Typesrings
; Artinian ring : A ring satisfyingdescending chain conditionleft idealsleft artinian; ifsatisfiesdescending chain conditionright ideals, itright artinian; if itboth leftright artinian, itcalled artinian''. Commutative artinian ringsnoetherian.
; Boolean ring : A ringwhich every elementidempotent isboolean ring.
; Commutative ring : A ring Rcommutative ifmultiplicationcommutative, i.e. rs=srall r,s∈R.
; Dedekind domain :
; Division ring or skew field : A ringwhich every nonzero element isunit1≠0 isdivision ring.
; Euclidean domain : An integral domainwhichdegree functiondefined so that "divisionremainder" can be carried outcalledEuclidean domain (becauseEuclidean algorithm worksthese rings). All Euclidean domainsprincipal ideal domains.
; Field : A commutative division ring isfield. Every finite division ring isfield, asevery finite integral domain. Field theoryindeed an older branchmathematics than ring theory.
; Integral domain : A commutative ring without zero divisorsin which 1≠0an integral domain.
; Local ring : A ring withunique maximal left ideal islocal ring. These rings also haveunique maximal right ideal, andleft andright unique maximal ideals coincide.
; Noetherian ring : A ring satisfyingascending chain conditionleft idealsleft noetherian;ring satisfyingascending chain conditionright idealsright noetherian;ring thatboth leftright noetheriannoetherian. A ringleft noetherian ifonly if all its left idealsfinitely generated; analogouslyright noetherian rings.
; Semi-simple ring :
; Simple ring : A ringno two-sided Ideals.
; Unique factorization domain :
; Principal ideal domain : An integral domainwhich every idealprincipal isprincipal ideal domain. All principal ideal domainsunique factorization domains.
Miscellaneous
; Direct productdirect sums : Thesewaysconstruct new rings from given ones; please refer tocorresponding linksexplanation.
; Krull dimension ofcommutative ring : The maximal length ofstrictly increasing chainprime ideals inring.
; Localization ofring : A techniqueturngiven setelements ofring into units in"best possible" way.
; Subring : A subset S ofring (R,*,+) which remainsring when +*restrictedScontainsmultiplicative identity 1RcalledsubringR.
; Rig : A rigan algebraic structure satisfyingsame properties asring, except that addition need only be an abelian monoid operation, rather than an abelian group. The term "rig"meantsuggest thatis"ring" without "n"egatives.
; Rng : A rngan algebraic structure satisfyingsame properties asring, except that multiplication need not have an identity element. The term "rng"meantsuggest thatis"ring" without an "i"dentity.
