查询词典 injective dimension

Finally, it investigate QF-ring with right gpm-injective, and obtain the following theorem:Theorem The following statements are equivalent:(1) R is quasi-Frobeniusean ring:(2) R is right noetherian, left pm-injective ring, right pm-injective ring;(3) R is right noetherian, left pm-injective ring, right GP-injective ring;(4) R is right noetherian, left gpm-injective ring, right gpm-injective ring;(5) R is right noetherian, right gpm-injective ring and each right minimal ideal is right annihilator.

最后用右gpm-内射对QF环进行研究，得到了如下定理：定理下列条件等价：(1) R是quasi-Frobeniusean环；(2) R是右Noether，左pm-内射环，右pm-内射环：(3) R是右Noether，左pm-内射环，右GP-内射环；(4) R是右Noether，左gpm-内射环，右gpm-内射环；(5) R是右Noether，右gpm-内射环且每个极小右理想是右零化子。

In this paper, we study syzygies of injective resolvent s and cosyzygies of injective resolutions, and probe the relationship of injective resolvent s and injective resolutions, and then prove that if R is a Noetherian ring with id≤n, then every left R-module has injective dimension at most n or ∞.

研究了内射预解式的合冲模与内射分解式的上合冲模，并探讨了内射预解式与内射分解式之间的联系，证明了如果环R是Noether环且id≤n，则每个左R-模的内射维数小于等于n或者为∞。

The injective rings play an important role in the study of rings and categories of modules . First , we introduce the notion of ann-injective rings and CT-injective modules .Second,we make an inquiry into a series of their properties.Third,we give the definition of homological dimension of CT-injective modules.At last,we give the definition of FGT rings which is the extension of cogenerator rings.

本文对环与模范畴中一重要的模类—内射模进行了延拓，引入了ann -自内射环以及CT -内射模的概念，探讨了它们一系列的性质，并定义了CT -内射模的同调维数，最后对余生成子环进行推广得到了FGT -环，讨论了它与CT -内射环的关系以及它的一些性质。

Next, we discuss the relations between left quasi-dual bimodules and left dual-bimodules, we obtain that a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions: sM is minimal injective and MR is a M-minimal injective kasch-module; MR is a M-minimal injective kasch-module and for any two ideals LI and L2 ofSS rM(L1 n L2)-rw(L1)+rM(I2); sM is minimal injective and for any two submodules A and B of MR,Lastly, we applicate the quasi-duality on smash product algebra R#H, and obtain an answer of the semiprime problem, i.e., let H be a finite-dimensional semisimple Hopf algebra and R be an H-module algebra, if R is left quasi-dual and semiprime, then R#H is semiprime.

我们得到：一个左拟对偶双边模如果满足下列条件之一，则它将成为一个左对偶双边模：_sM是单内射的并且M_R是一个M-单内射kasch-模；M_R是一个M-单内射kasch-模并且对_sS的任意两个理想，有r_M(L_1∩L_2)=r_M(L_1)+r_M(L_2)；_sM是单内射的且对M_R的任意两个子模，有l_s=l_s+l_s。2 在第2.3节中我们将拟对偶性应用于smash积代数R#H，部分解决了半素问题。

Soc _RR is a injective left R-module.(4) Soc _RR is a left self-injective ring.|Theorem 4.2.9 Let R be a commutative notherian ring, then the following are equivalent:(1) R is a FS-ring.(2) R is a PS-ring.(3) Every minimal ideal of R is injective.(4) Every minimal ideal of R is p-injective.

定理4.2.9若R是交换的Noethe(来源：Aae0eBe0C论文网www.abclunwen.com)r环，则以下条件等价：R是FS-环；R是PS-环；R的每个极小理想都是内射的；R的每个极小理想都是p-内射的；R是IS-环；R的每个极小理想都是投射的；R的每个极小理想都是平坦的。

Theorem 2.1.4 The following statements are equivalent for a ring R:(1) R is left co-semihereditary;(2) every finitely cogenerated factor module of a finitely cogenerated injective R-module is quasi-injective;(3) every finitely cogenerated factor module of a finitely cogenerated injective R-module is direct-injective.

an环 R下列叙述等价： l）R的每个商环是强左余半遗传的； 2）R的每个商环是左余半遗传的； 3）拟内射R模的每个有限余生成商模是拟内射的； 4）有限余生成拟内射R－模的每个有限余生成商是直内射的； 5）有限余生成fi－(来源：A3fBC论文网www.abclunwen.com)拟内射R－模的有限余生成商模是拟内射的。

In this paper, we further research P-injective modules, define the concept of prime injective modules and prime injective dimension of module, and discuss the properties of them.

本文在[3]的基础上进一步研究了P-内射环，对于交换环我们引入了素内射模和模的素内射维数的概念，讨论了它们的性质。

Firstly, an equivalence proposition of right qpm-injective module is discussed, and when the endomorphism of right qpm-injective module is right pm-injective ring is observed.

首先，讨论右qpm-内射模的一个等价命题，以及在什么情况下，右qpm-内射模的自同态环是右pm-内射环。

If M is self-generator, then the endomorphism of right qpm-injective module M is right pm-injective ring. Finally, the properties of the endomorphism of right qpm-injective module are discussed and obtain some applications.

且仅当如果ker=ker，其中s，t∈S=End，那么Ss=S_t；(2)如果右R-模M是自生成子，那么右qpm-内射模M的自同态环是右pm-内射环。

The first chapter, main instead " duo-ring " condition of " every maximal left ideal is GW-ideal " condition,study strongly regularities of GP-V-ring on this condition.lt is shown that (1) R is strongly regular iff R is left GP-V-ring whose maximal left ideals are GW-ideal.(2)R is strongly regular iff R is left GP-V-ring whose maximal right ideals are GW-ideal. The second chapter, generalize some results of GP-V-ring to GP-V-ring, discuss regularity of GP-V-ring.It is shown that (1) R is left self-injective regular with non-zero socle iff R is left GP-V -ring with Soc = Soc and R contains an injective maximal left ideal.(2)R is regular ring and every maximal essential left ideal is ideal iff R is left GP-injective left GP-V -ring and every maximal essential left ideal is ideal.

第一章主要将"duo-环"条件替换成"每一极大左理想是GW-理想"条件，研究在此条件下，GP-V-环的强正则性，证明了：(1)R是强正则环当且仅当R是左GP-V-环且R的每一极大左理想是广义弱理想；(2)R是强正则环当且仅当R是左GP-V-环且R的每一极大右理想是广义弱理想，第二章，主要将GP-V-环上一些结果推广到GP-V′-环上，讨论GP-V′-环的正则性，证明了：(1)R是左自内射正则环且Soc≠0当且仅当R是包含内射极大左理想的GP-V′-环，且Soc=Soc；(2)R是正则环且每一极大本质左理想是理想当且仅当R是左GP-内射的左GP-V′-环且每一极大本质左理想是理想。

This matter overthrew all basises I learned from Sunday school .

这件事把我从周日学校学到的所有道理都颠覆了。

It's one of the cruel ironies of aging: if you want to remember your favorite restaurants, the less you should eat at them.

这是一个老化的残酷讽刺：如果你想记住你最喜欢的餐馆，你应该更少吃到它们。