半质环的中心元

Central Elements of Semiprime Rings

本文给出了半质环的元为中心元的条件。

Let Ω be a semiprime ring with centre Z, a∈Ω. If Ωsatisfies one of the following conditions: (1) for all x∈Ω, (ax)~2—x^2a^2∈Z, (2) for all x ∈Ω, (xa)~2—a^2x^2∈Z, then a∈Z. If a≠0, Ωis a 2-torsionfree prime ring and satisfies one of the following conditions: (1) for all x ∈Ω, (ax)~2+(xa)~2∈Z, (2) for all x∈Ω, ax^2a+xa^2x∈Z, then Ω is commutative. IfΩsatisfies one of the following conditions: (1) for all x,y∈Ω, [x^2y^2+y^2x^2,x]=0, (2) for all x, y∈ Ω,[(xy)~2+(yx)~2,x]=0, (3) for all x,y∈Ω,[(xy)~2—y^2x^2,x]=0,(4) for all x,y∈Ω, [(xy)~2—y^2x^2,y]=0, (5) for all x, y∈Ω,[xy^2x+yx^2y,x]=0, (6) for all x,y∈Ω,[(xy)~2— x^2y^2,x]=0, (7) for all x, y∈Ω,[(xy)~2—x^2y^2,y]=0, then Ω is commutative.