理想的不可约性与可消性

发布时间：2018-06-14 06:04

  本文选题：正则环 + 可消理想　； 参考：《南京理工大学》2017年硕士论文

【摘要】：完全强不可约理想是强不可约理想在[1]从有限到无限的一种自然推广,也是完全不可约理想在[2]的强化。本文通过建立完全强不可约理想的刻画,考察完全强不可约理想与素理想的关系,以及完全强不可约理想与完全不可约理想的关系,得出:R是一个环且J(R)= 0,则每个完全不可约理想是完全强不可约的当且仅当R是既是正则的,又是半完全的。这一关系导致产生了一种新的环类——完全算术环。与此同时,在正则环的基础上,进一步考察了这种环的结构。文章的第二部分,我们主要研究理想的可消性。理想的可消性是交换代数学研究的一个十分活跃的课题。对于正则算术环,我们建立了可消理想的一系列刻画,为进一步研究相应环类中理想的可消性提供了一定的基础。作为这一结果的应用,我们可以更清晰地了解完全算术环中理想的可消性,并给出完全算术环中可消理想的等价条件:R是一个完全算术环且J(R)=0。那么N是一个可消理想当且仅当对于任意e∈Idem(R),存在f ∈Idem(N)使得Re=Rf。
[Abstract]:A completely strongly irreducible ideal is a natural generalization of a strongly irreducible ideal from finite to infinite in [1], and is also a reinforcement of completely irreducible ideal in [2]. In this paper, the relations between completely strongly irreducible ideals and prime ideals, and between completely strongly irreducible ideals and completely irreducible ideals are investigated by establishing a characterization of completely strongly irreducible ideals. In this paper, we obtain that: r is a ring and J n R is 0, then every completely irreducible ideal is completely irreducible if and only if R is both regular and semi-complete. This relation leads to a new class of rings-complete arithmetic rings. At the same time, on the basis of regular rings, the structure of these rings is further investigated. In the second part of the paper, we mainly study the ideality. Ideal cancellability is a very active subject in the study of commutative algebra. For regular arithmetic rings, we establish a series of characterizations of cancelable ideals, which provide a basis for further study of the cancellability of ideals in the corresponding ring classes. As an application of this result, we can understand the cancelability of ideals in complete arithmetic rings more clearly, and give the equivalent condition of vanishing ideals in complete arithmetic rings: R is a complete arithmetic ring and J ~ + R ~ (0). Then N is a cancelable ideal if and only if, for any e 鈭，

本文编号：2016385