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This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.
      
TheS-arithmetic group г is of typeFn, resp.FPn, if and only if for allp inS thep-adic completionGp of the corresponding algebraic groupG is of typeCn resp.CPn.
      
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We next show that this result is best possible by including a result of Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis.
      
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In this paper, we investigate more general rings than PF rings, called fPF rings.We prove the following results:(1) R is a fPF ring if and only if for each a∈R there exists f∈H(R) such that ann_R(f(a)) is a pure ideal of R. (2) Let R be a local ring. Then R is a fPF ring if and only if for every a∈R, there exists f∈H(R) such that f(a) is a non zero divisor or f(a)=0. (3) Ring R is a fPF ring if and only if for every a∈R, there is f∈H(R) such that f(a) is a non zero divisor in each...

In this paper, we investigate more general rings than PF rings, called fPF rings.We prove the following results:(1) R is a fPF ring if and only if for each a∈R there exists f∈H(R) such that ann_R(f(a)) is a pure ideal of R. (2) Let R be a local ring. Then R is a fPF ring if and only if for every a∈R, there exists f∈H(R) such that f(a) is a non zero divisor or f(a)=0. (3) Ring R is a fPF ring if and only if for every a∈R, there is f∈H(R) such that f(a) is a non zero divisor in each localization Rp or f(a) =0 in each Rp. (4) Ring is a strongly fPF ring, for x∈R, a∈ann_R (x) if and only if f(a)∈ann_R(x), then R is a PF ring. We also provide an example of fPF ring which is not a PF ring.

本文研究了较PF—环更广泛的一类环,被称为fPF—环,得到如下结果:(1)R是fPF—环当且仅当对于每个a∈R,存在f∈H(R),使得ann_R(f(a))是R的一个纯理想;(2)设R是局部环,则R是fPF—环当且仅当对于每个a∈R,存在f∈H(R)使得f(a)=0或者f(a)不是零因子;(3)R是fPF—环当且仅当对每个a∈R。存在f∈H(R)使得f(a)在每个局部化Rp中不是零因子,或者在每个Rp中f(a)=0;(4)设R是强fPF—环,且对于x∈R,a∈ann_R(x)当且仅当f(a)∈ann_R(x)对某f∈H(R),则R是PF—环,另外,我们还给出一个例子说明fPF—环是PF—环的严格推广

In this paper. we prove the following results: (1) A ring R is strongly regular if and only if R is a left p-V-ring and verey maximal left ideal of R is a quasiideal; (2) A ring R is trongly regular if and only if R is fully semiprime and every maximal left subideal in a principal ideal of R is an ideal. Therefore. the following important theorem of Kaplansky is effectively extended: A commutative ring R is von Neumann regular if and only if every simple R -module is injective. Further properties...

In this paper. we prove the following results: (1) A ring R is strongly regular if and only if R is a left p-V-ring and verey maximal left ideal of R is a quasiideal; (2) A ring R is trongly regular if and only if R is fully semiprime and every maximal left subideal in a principal ideal of R is an ideal. Therefore. the following important theorem of Kaplansky is effectively extended: A commutative ring R is von Neumann regular if and only if every simple R -module is injective. Further properties of p-V-rings are also considered .

本文中,我们证明了如下结果:(1)环R是强正则的当且仅当R是左P-V-环且R的每个极大左理想是拟理想;(2)环R是强正则的当且仅当R是半素的且R的主左理想的极大左次理想是R的理想,所以有效推广了Kaplansky的如下结果:可换环R是VonNeumann正则的当且仅当每一个单R-模是内射的。

in the paper, we proved the problem of (2),and proved that R is r-torsionregular ring if and only if R is tv-torsion ring.

本文证明了文献[2]提出的问题,并且更进一步的证明了环R是τV-挠正则环当且仅当R是,τV-挠环.

 
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