线性代数网课代修|同调代数代写homological algebra代考|MATH813

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

线性代数作业代写linear algebra代考|Hereditary and Dedekind Rings

We have seen that assuming every $R$-module is “special” (projective, injective, or flat) constrains $R$. Moreover, interesting rings are characterized in this way. We now assume that every ideal is special.
Assuming that every left ideal is injective gives nothing new.

Proposition 4.11. Every left ideal in a ring $R$ is injective if and only if $R$ is semisimple.

Proof. The submodules of $R$ are its left ideals. As each left ideal is injective, it is a direct summand, by Corollary 3.27. Proposition $4.1$ now says that $R$ is a semisimple left $R$-module; that is, $R$ is a (left) semisimple ring. Conversely, if $R$ is semisimple, then every left ideal is injective, by Proposition 4.5.

Definition. A ring $R$ is left hereditary if every left ideal is projective; a ring $R$ is right hereditary if every right ideal is projective. A Dedekind ring is a hereditary domain.

线性代数作业代写linear algebra代考|Semihereditary and Prufer Rings

We now investigate rings in which all finitely generated ideals are special.
Definition. A ring $R$ is left semihereditary if every finitely generated left ideal is projective. A semihereditary domain is called a Prüfer ring.
Example 4.27.
(i) Every left hereditary ring is left semihereditary (of course, these notions coincide for left noetherian rings).
(ii) Chase gave an example of a left semihereditary ring that is not right semihereditary (see Lam, Lectures on Modules and Rings, p. 47). A theorem of Small says that a one-sided noetherian ring is left semihereditary if and only if it is right semihereditary (see Lam, p. 268).
(iii) Every von Neumann regular ring is both left and right semihereditary. By Lemma $4.8$, every finitely generated left (or right) ideal $I$ is principal; say, $I=(a)$. If $a a^{\prime} a=a$, the map $\varphi: R \rightarrow I$, defined by $\varphi(r)=r a^{\prime} a$, is a retraction. Therefore, $I$ is a direct summand of $R$ and, hence, $I$ is projective.

Definition. A ring $R$ is a Bézout ring if it is a domain in which every finitely generated ideal is principal.

It is clear that every Bèzout ring is a Yrüfer rıng; 1.e., it is semmereditary.

线性代数作业代写linear algebra代考|Semihereditary and Prufer Rings

Prüfer 环。

(i) 每个左遗传环都是左半遗传的（当然，这些概念与左诺特环一致）。
(ii) Chase 举了一个左半遗传环不是右半遗传的例子（参见 Lam, Lectures on Modules and Rings, p. 47) 。Small 的一个定理说，一个单侧的诺特环是左半遗传的当且仅当它是右半 遗传的（参见 Lam, p. 268）。
(iii) 每个冯诺依曼正则环都是左右半遗传的。引理 $4.8$, 每个有限生成的左（或右）理想 $I$ 是 校长；说， $I=(a)$. 如果 $a a^{\prime} a=a$ ，地图 $\varphi: R \rightarrow I$ ，被定义为 $\varphi(r)=r a^{\prime} a$, 是撤回。所 以， $I$ 是直接总和 $R$ 因此， $I$ 是投射的。

计量经济学代写

在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions