The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- reg...The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.展开更多
Initially, Osgood used the integral ∫dr/f（r）for an unicity crite, rion to the differential equation y＇ = f（y）, f （0） = 0. The trivial solution is unique iff this integral goes to the infinite at the origin. Th...Initially, Osgood used the integral ∫dr/f（r）for an unicity crite, rion to the differential equation y＇ = f（y）, f （0） = 0. The trivial solution is unique iff this integral goes to the infinite at the origin. Then he can prove the unicity of the trivial solution of y＇ = y Ln｜Y｜, although the second member is not lipschitzian. Later, Bernfeld [1] shows that all the solutions of y＇ = f（y） do not explose iffthe same integral goes to the infinite at the infinite. Finally, we can adapt a result from the Cauchy works as follows： the trivial solution is a singular solution iffthe same integral vanishes at the origin. Using non standard analysis, we present the proofs of the different criterions and show that the Osgood integral was used by Cauchy before in the similar purpose.展开更多
基金The Foundation for Excellent Doctoral Dissertationof Southeast University (NoYBJJ0507)the National Natural ScienceFoundation of China (No10571026)the Natural Science Foundation ofJiangsu Province (NoBK2005207)
文摘The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.
文摘Initially, Osgood used the integral ∫dr/f（r）for an unicity crite, rion to the differential equation y＇ = f（y）, f （0） = 0. The trivial solution is unique iff this integral goes to the infinite at the origin. Then he can prove the unicity of the trivial solution of y＇ = y Ln｜Y｜, although the second member is not lipschitzian. Later, Bernfeld [1] shows that all the solutions of y＇ = f（y） do not explose iffthe same integral goes to the infinite at the infinite. Finally, we can adapt a result from the Cauchy works as follows： the trivial solution is a singular solution iffthe same integral vanishes at the origin. Using non standard analysis, we present the proofs of the different criterions and show that the Osgood integral was used by Cauchy before in the similar purpose.
