The Structure of Symplectic Groups over Arbitrary Commutative Rings

<正> Let R be an arbitrary commutative ring, and n be an integer≥3. It is proved for anyideal J of R thatESp2n（R,J）=[ESp2n（R）, ESp2n（J）]=[ESp2n（R）, ESp2n（R,J）] =[ESp2n（R）, GSp2n（R,J）]=[Sp2n（R）, ESp2n（R,J）].Furthermore, the problem of normal subgroups of Sp2n（R） has an affirmative solution if and only if aR=a2R+2aR for each a in R. This covers the relevant results of [4],[8],[10],[12] and [13].