發(fā)布時(shí)間：2018-07-05 07:59

  本文選題：GWCN環(huán) + 約化環(huán)�。� 參考：《安徽師范大學(xué)》2017年碩士論文

【摘要】：本文主要研究GWCN環(huán)的性質(zhì),討論了它推廣.通過(guò)研究GWCN環(huán),一方面發(fā)現(xiàn)了它與一些特殊環(huán)的關(guān)系,另一方面給出了其正則性及clean性,研究了 GWCN環(huán)的擴(kuò)張及推廣.首先,我們介紹GWCN環(huán)和其它環(huán)的關(guān)系,并且構(gòu)造若干反例說(shuō)明{ CN環(huán)}(?){GWCN環(huán)} (?) {弱半交換環(huán)},得到了 GWCN環(huán)成為約化環(huán)的條件.接著討論GWCN環(huán)的擴(kuò)張,如矩陣擴(kuò)張、局部化等.其次,研究了 GWCN環(huán)的正則性,證明了:(1)環(huán)R為約化環(huán)當(dāng)且僅當(dāng)R為CN環(huán)且R是n-正則的當(dāng)且僅當(dāng)R為GWCN環(huán)且R是n-正則的;(2)若R為GWCN環(huán),則R為左弱正則環(huán)當(dāng)且僅當(dāng)R為弱正則環(huán)和雙正則環(huán);(3)設(shè)R是有Abelian極大左理想的GWCN環(huán),則下列條件等價(jià):(a) 是強(qiáng)正則環(huán);(b) 為左GP-V'-環(huán),其極大本質(zhì)左理想均為廣義弱理想;(c) R是左GP-V'-環(huán),其極大本質(zhì)右理想均為廣義弱理想.討論了 GWCN環(huán)的clean性,證明了:設(shè)R為GWCN環(huán),則R為弱exchange環(huán)當(dāng)且僅當(dāng)R為弱clean環(huán).最后,提出了 GWCN環(huán)的推廣-α-GWCN環(huán),討論了它與一些特殊環(huán)的關(guān)系,研究了 α-GWCN環(huán)的一些性質(zhì),給出了:若α是環(huán)R的自同態(tài),I為R的理想,α(I) (?) I,那么:(1)若I (?)N(R),R為α-GWCN 環(huán),那么R/I 為α-GWCN環(huán);(2)若I是約化的,且R/I為α-GWCN環(huán),那么R為α-GWCN環(huán).其中α::R/I →R/I,α(a + I) = α(a) + I,任意 a ∈ R.
[Abstract]:In this paper, we study the properties of GWCN ring and discuss its generalization. By studying GWCN rings, the relations between GWCN rings and some special rings are found, on the other hand, the regularity and clean properties of GWCN rings are given, and the extension and generalization of GWCN rings are studied. First, we introduce the relations between GWCN rings and other rings, and construct some counterexample to explain {CN rings} (?) {GWCN rings} (?) {weakly semicommutative rings}, and obtain the conditions under which GWCN rings become reduced rings. Then we discuss the extension of GWCN rings, such as matrix extension, localization and so on. Secondly, we study the regularity of GWCN rings and prove that: (1) A ring R is a reduced ring if and only if R is a CN ring and R is n- regular if and only if R is a GWCN ring and R is n- regular; (2) if R is a GWCN ring, Then R is a left weakly regular ring if and only if R is a weakly regular ring and a biregular ring. (3) Let R be a GW CN ring with a Abelian maximal left ideal, then the following conditions are equivalent: (a) is a strongly regular ring; (b) is a left GP-Va-ring. The maximal essential left ideals are all generalized weak ideals; (c) R are left GP-Va-rings, and the maximal essential right ideals are all generalized weak ideals. In this paper, we discuss the clean property of GWCN rings, and prove that if R is a GWCN ring, then R is a weak exchange ring if and only if R is a weak clean ring. Finally, the generalized 偽 -GWCN ring of GWCN ring is proposed, the relation between 偽 -GWCN ring and some special rings is discussed, some properties of 偽 -GWCN ring are studied, and the following results are given: if 偽 is the ideal of the endomorphism of ring R, 偽 (I) (?) Then: (1) if I (?) N (R) R is 偽 -GWCN ring, then R / I is 偽 -GWCN ring; (2) if I is reduced and R / I is 偽 -GWCN ring, then R is 偽 -GWCN ring. Where 偽: r / R / I / R / I, 偽 (a I) = 偽 (a) I, any a 鈭，

本文編號(hào)：2099535