投射模

Schauengurg[2] considered more general case and found necessary and sufficient conditions for the cosmash product coalgebra H Q to be a Hopf algebra with a Hopf subalgebra H and a weak projection !

Schauenburg在文献[2]中把Radford的研究加以推广，给出了使余smash积HQ为Hopf代数，H为HQ的子Hopf代数，且具有弱投射π：HQ→H(即π是左模余代数同态，且π|_H=id_H)的等价条件。

Then we introduce the concept of(n,0)-projective dimension for modules and rings.It measures how far away a finitely generated module is from being n-presented,and how far away a ring is from being Noetherian.

然后，我们介绍环与模的(n，0)-投射维数，它可以衡量一个有限生成模与n-表现模的差距，以及衡量一个环与Noether环的差距。

In section one, we study regular cu-rings by use of full elements in rings, and give some equivalent conditions for the endomorphism ring of quasi-projective modules and quasi-injective modules to be a cu-ring.

在第一部分，我们利用环中的完全元对正则cu-环进行刻画，并给出了拟投射模和拟内射模的自同态环是cu-环的一系列等价条件。

Finally, sufficient and necessary conditions for endomorphism rings of some special modules to be semi-π-regular are given.

最后，我们对半π正则环作了一定的研究，给出了一个直投射模的自同态环是半π正则环的充要条件。

Due to the inspiration of maximal injective module by professor Wang Ming-yi, in chapter two, we introduce the conception of max-projective modules, discuss its properties, applications: the primitive relations between rings and endomorphism rings, an equivalence that is surrounding strongly V.N. rings.

受汪明义教授引进极大内射模概念的启发，在第二章中我们引进了极大投射模的概念，由此讨论了它的一些性质及其应用：环与自同态环之间的本原性关系；强正则环的一个等价刻划；并在环为交换和非交换的情形下分别应用有限表现模的极大投射性得出了SF环和Von Neumann正则环的关系。

Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.

最后，作为应用，证明了每一纯投射左R-模在纯子模下封闭当且仅当每一纯内射左R-模在纯满像下封闭。

Every module is a submodule an injective module and a homomorphic image of a projective module.

任何模都是内射模的子模，也是投射模的同态像。

The innovation of our thesis lies in:1 We do not require quadratic space to have hyperbolic basis. Moreover, we only require M to be projection module instead of free module;2 Eichler transformation is used instead of the traditional matrix transformation in our proof;3 Not only do we extend the relative results obtained by Bak and Vavilov and the results obtained by Vaserstein and You hong , we also simplify the proof significantly.

我们的创新之处在于： 1 不要求二次型空间具有双曲基，甚至不要求M是自由模，而仅要求M是投射模； 2 我们在论证中使用的是Eichler变换而不是传统的矩阵； 3 我们的结果不仅推广了Bak与Vavilov以及Vaserstein与游宏的相应结果，而且论证也极大地简化了。

Then we discuss comodule algebra on the Hopf algebroid. If L is finite generated as a left R-projective module, left L-comodule category is equivalent to the left L〓-module category.

然后讨论Hopf代数胚的余模代数，如果L是有限生成的左R-投射模，那么左L-余模范畴与左L〓-模范畴是等价的。

In chapter 4, we define the projective dimension of flat modules, use it to characterize many rings, and the relations between cotorsion modules and the projective dimension of flat modules are also given.

在第四章中，我们定义了平坦模的投射维数，用它刻划了一些环，并讨论了Cotorsion模和严坦模的投射维数的关系。

With Death guitarist Schuldiner adopting vocal duties, the band made a major impact on the scene.

随着死亡的吉他手Schuldiner接受主唱的职务，乐队在现实中树立了重要的影响。

But he could still end up breakfasting on Swiss-government issue muesli because all six are accused of nicking around 45 million pounds they should have paid to FIFA.

不过他最后仍有可能沦为瑞士政府&议事餐桌&上的一道早餐，因为这所有六个人都被指控把本应支付给国际足联的大约4500万英镑骗了个精光。