P-内射的WB-环

研究了满足一定条件的P-内射环为WB-环的等价刻画.证明了如果R是非奇异的P-内射环,那么R只要满足条件之一:(a)R满足特殊左零化子的升链条件;(b)R不包含由有限非零主左理想构成的直和项;(c)R是CF环;(d)R是Goldie环.有如下等价:(1)R是WB-环;(2)对任何a∈R,有正交理想I,J,使得a=aua=ava,这里u∈R,模I右可逆,v∈R模J左可逆;(3)对任何a∈R,有正交理想I,J和幂等元e∈R,使得a=eu=ev,这里u∈R模I右可逆,v∈R模J左可逆;(4)如果ab,a,b∈R,则有正交理想I,J,使得au=ub,av=vb,其中u∈R模I右可逆,v∈R模J左可逆.

Principal-injective WB-rings

Necessary and sufficient conditions under which a principal injective ring＇s equivalence is a WB-ring. It is proved that if R is a nonsingular and principal ring, R just satisfies one of the following con- ditions： （a） R satisfies the ascending chain condition for special left annihilators; （b） R does not contain a direct sum of an infinite number of non-zero principal left ideals; （c） R is a CF-ring; （d） R is a Goldie ring, then the following conditions are equivalent： （1） R is a WB-ring; （2） For any a E R ,there exists or- thogonal ideals I and J such that a = aua = ava, where u ∈R is fight invertible module I and v E R is left invertible module I; （3） For any a E R, there exists orthogonal ideals I, J and e = e^2∈ R such that a = eu = ev,whereu∈R is right invertible module I and v∈R is left invertible module J;（4） If a=b, a, b ∈ R, then there exists orthogonal ideals I, J such that au = ub, av = vb, where u E R is right invert- ible module I and v E R is left invertible module J.