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IdealCommutative ringPrincipal idealJacobson radicalIdempotentPrime elementRing homomorphismQuotient ringRing theoryCommutative algebraMaximal idealIdeal class groupConductorIntegral domainMatrix ringMonoidRoot of unity modulo nConjugacy classInitial and terminal objectsFormal derivativeNested radicalCommutator subgroupCharacteristicElementEuclidean domainGraded ringAlgebraic elementAnnihilatorFree algebraNoncommutative ringProductGeneralized inverseRinged spaceAlgebraic numberGenerating set of a groupRational numberAscending chain conditionSum of two squares theoremInvariant basis numberUnique factorization domainInner automorphismRadical symbolDedekind domainInclusion mapAntiderivativeRing of integersRank factorizationPolynomial ringRational functionRational root theoremRingClass field theoryLinear extensionLie algebra representationClassFundamental representationRepresentation theoryInvariant theoryComposition algebraLimitUpper and lower boundsAlgebraic number theoryNoetherian ringCyclotomic fieldAlgebraic expressionLie algebra extensionSpectrum of a ringPrimary idealCharacter tableHermitian adjointMatrix pencilFieldEquivalence of categoriesIncidence algebraPrimary decompositionPrimitive element theoremNatural transformationArtinian ringGroupUniversal propertyAxiom of extensionalityRepresentation theoremLatticeComplex numberArchimedean propertyIntersection theoryGroup isomorphismAlgebraically closed fieldPerfect powerDiagramCharacter theoryCongruence relationBandSymmetric groupChain ruleCompact operatorIndicator functionDualityPolarization identityIrreducible element