满足SR2*(R,I)条件的环的相对K2

设I是环R的理想,称(R,I)满足SR_2~*主(R,I)条件,如果它满足SR_2(R,I)条件,并且对任意的a,b∈I,存在一个I-单位半正则元t∈R,使得1 a(b-t)∈U(R,I),称环R带许多单位半正则元,如果它满足SR_2~*(R,R)条件,本文证明,如果(R,I)满足SR_2~*(R,I)条件,则S(R,I)=L(I)(?)(I)L(I)H(R,I),且相对K′_2群K′_2(R,I)(K_2(R,I))包含在H(R,I)((?)(R⊕I,O⊕I))中；进而,若I包含于R的中心,则K′_2(R,I)和K_(R,I)由相对Dennis-Stein符号生成,特别的,如果R是带许多单位半正则元的环,那么K_2(R)包含在H(R)中；进而,若R是交换的,则K_(2(R)由Dennis-Stein符号生成,在SR_2(R,I)条件下,本文证明了K_2(n,R,I)具有满稳定性,其中n≥3。

Relative K2 of Rings With SR2*(R,I) Condition

Let R be a ring and I an ideal of R.(R,I)is said to satisfy the SR2*(R,I)condition if the SR2(R,I)condition is satisfied, and for any a, b ∈ I, there exists anI-unit semi-regular element t ∈ R such that 1+a(b-t)∈ U(R,I). A ring R is said to have many unit semi-regular elements if the SR2*(R,R)condition is satisfied. It is proved in this paper that if the SR2*(R,I)condition is satisfied, then S(R,I)= L(I)( )(I)L(I)H(R,I)and the relative K2 group K12(R,I)(K2(R,I))is contained in H(R,I)(H(R( )I,O( )I)); and if furthe R,I is contained in the center of R, then K12(R,I)and K2(R,I)are generated by the relative Dennis-Stein symbols. In particula R,I f R has many unit semi-regular elements, then K2(R)is contained in H(R);and if further, R is commutative, then K2(R)is generated by the Dennis Stein symbols. The surjective stability of K2(n,R,I)is also obtained under the SR2(R,I)condition and n ≥ 3.