本文選題：正則環(huán) + 可消理想��； 參考：《南京理工大學(xué)》2017年碩士論文

【摘要】：完全強不可約理想是強不可約理想在[1]從有限到無限的一種自然推廣,也是完全不可約理想在[2]的強化。本文通過建立完全強不可約理想的刻畫,考察完全強不可約理想與素理想的關(guān)系,以及完全強不可約理想與完全不可約理想的關(guān)系,得出:R是一個環(huán)且J(R)= 0,則每個完全不可約理想是完全強不可約的當且僅當R是既是正則的,又是半完全的。這一關(guān)系導(dǎo)致產(chǎn)生了一種新的環(huán)類——完全算術(shù)環(huán)。與此同時,在正則環(huán)的基礎(chǔ)上,進一步考察了這種環(huán)的結(jié)構(gòu)。文章的第二部分,我們主要研究理想的可消性。理想的可消性是交換代數(shù)學(xué)研究的一個十分活躍的課題。對于正則算術(shù)環(huán),我們建立了可消理想的一系列刻畫,為進一步研究相應(yīng)環(huán)類中理想的可消性提供了一定的基礎(chǔ)。作為這一結(jié)果的應(yīng)用,我們可以更清晰地了解完全算術(shù)環(huán)中理想的可消性,并給出完全算術(shù)環(huán)中可消理想的等價條件:R是一個完全算術(shù)環(huán)且J(R)=0。那么N是一個可消理想當且僅當對于任意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，

本文編號：2016385