【作者】 田艳；

【导师】 王芳贵；

【作者基本信息】 四川师范大学 ， 基础数学， 2009， 硕士

【摘要】 本文首先在交换环中引入gυ-理想、gυ-无挠模、ω~T-模,模的ω~T-包络;给出了ω~T-Noether环的定义.我们讨论了ω~T-Noether环的相伴素理想和准素分解.通过引入ω~T-不可约理想,证明了ω~T-Noether环上的任何真ω~T-理想都有准素ω~T-理想的准素分解.通过研究ω~T-Noether环上内射模的结构和性质,将Cartan-Eilenberg-Bass定理推广到了ω~T-Noether环上: R是ω~T-Noether环当且仅当任意多个gυ-无挠内射模的直和是内射模,当且仅当每个gυ-无挠内射模是Σ-内射模.我们还证明了若R是ω~T-Noether环,I是真ω~T-理想,则E(R/I)可以写成有限个不可分解内射模的直和.最后,我们引入了υ-Noether环的定义(具有υ-理想的升链条件的交换环).证明了每个ω~T-Noether环都是υ-Noether环;证明了若P为R中的素理想, R是υ-Noether环,则R[P]也是υ-Noether环;还证明了R中每个非零理想都被包含在至多有限个极大t-理想中,如果对于每个极大t-理想M而言, R[M]是υ-Noether环,则R是υ-Noether环.更多还原

【Abstract】 In this paper, we introduce gυ-ideal, gυ-torsion-free module,ω~T-module,ω~T-envelope in Commutative rings and defineω~T-Noetherian ring. We discuss as-sociate prime ideals and primary decomposition ofω~T-Noetherian rings. Byintroducingω~T-irreducible ideal, we prove that any properω~T-ideal has primarydecomposition of primaryω~T-ideal. By studying the structure and propertiesof injective modules ofω~T-Noetherian rings, we extend Cartan-Eilenberg-BassTheorem toω~T-Noetherian rings: R is aω~T-Noetherian ring if and only if everydirect sum of gv -torsion-free injective modules is injective; if and only if everygv -torsion- free injective module isΣ-injective. Let R be aω~T-Noetherian ring,we also prove that E(R/I) can be presented finite direct sum of indecomposableinjective modules if I is a properω~T-ideal. Finally, we defineυ-Noetherian ring(ie. having the condition of ascending chain onυ-ideals). We prove that everyω~T-Noetherian ring is aυ-Noetherian ring. Let R be aυ-Noetherian ring, weprove that R[P] is aυ-Noetherian ring if P is a prime ideal. We also prove that ifeach nonzero ideal is contained in at most finitely many maximal t-ideals and foreach maximal t-ideal M, R[M] is aυ-Noetherian ring, then R is aυ-Noetherianring.更多还原