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单词 ProofOfCharacterizationsOfTheJacobsonRadical
释义

proof of characterizations of the Jacobson radical


First, note that by definition a left primitive ideal is the annihilatorMathworldPlanetmath of an irreducible left R-module, so clearly characterizationMathworldPlanetmath 1) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the definition of the Jacobson radicalMathworldPlanetmath.

Next, we will prove cyclical containment. Observe that 5) follows after the equivalence of 1) - 4) is established, since 4) is independent of the choice of left or right idealsMathworldPlanetmathPlanetmath.

  1. 1) 2)

    We know that every left primitive ideal is thelargest ideal contained in a maximal left ideal. So the intersectionMathworldPlanetmathof all left primitive ideals will be contained in the intersection of allmaximal left ideals.

  2. 2) 3)

    LetS={M:M a maximal left ideal of R} and take rR.Let tMSM. Then rtMSM.

    Assume 1-rt is not left invertible; therefore there exists a maximalleft ideal M0 of R such that R(1-rt)M0.

    Note then that 1-rtM0. Also, by definition of t, we havertM0. Therefore 1M0; this contradictionMathworldPlanetmathPlanetmath implies 1-rt isleft invertible.

  3. 3) 4)

    We claim that 3) satisfies the condition of 4).

    Let K={tR:1-rt is left invertible for all rR}.

    We shall first show that K is an ideal.

    Clearly if tK, then rtK.If t1,t2K, then

    1-r(t1+t2)=(1-rt1)-rt2

    Now there exists u1 such that u1(1-rt1)=1, hence

    u1((1-rt1)-rt2)=1-u1rt2

    Similarly, there exists u2 such that u2(1-u1rt2)=1, therefore

    u2u1(1-r(t1+t2))=1

    Hence t1+t2K.

    Now if tK,rR, to show that trK it suffices toshow that 1-tr is left invertible. Suppose u(1-rt)=1, henceu-urt=1, then tur-turtr=tr.

    So (1+tur)(1-tr)=1+tur-tr-turtr=1.

    Therefore K is an ideal.

    Now let vK. Then there exists u such that u(1-v)=1, hence1-u=-uvK, so u=1-(1-u) is left invertible.

    So there exists w such that wu=1, hence wu(1-v)=w, then 1-v=w.Thus (1-v)u=1 and therefore 1-v is a unit.

    Let J be the largest ideal such that, for all vJ, 1-v is aunit. We claim that KJ.

    Suppose this were not true; in this case K+J strictly contains J.Consider rx+syK+J with xK,yJ and r,sR.Now 1-(rx+sy)=(1-rx)-sy, and since rxK, then 1-rx=u for someunit uR.

    So 1-(rx+sy)=u-sy=u(1-u-1sy), and clearly u-1syJ sinceyJ. Hence 1-u-1sy is also a unit, and thus 1-(rx+sy) isa unit.

    Thus 1-v is a unit for all vK+J. But this contradicts theassumptionPlanetmathPlanetmath that J is the largest such ideal. So we must haveKJ.

  4. 4) 1)

    We must show that if I is an ideal such thatfor all uI, 1-u is a unit, thenIann(MR) for every irreducible leftR-module MR.

    Suppose this is not the case, so there exists MR such thatIann(MR). Now we know thatann(MR) is the largest ideal inside some maximalleft ideal J of R. Thus we must also have IJ,or else this would contradict the maximality ofann(MR) inside J.

    But since IJ, then by maximality I+J=R, hence thereexist uI and vJ such that u+v=1. Then v=1-u, sov is a unit and J=R. But since J is a proper left ideal, thisis a contradiction.

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