Google 翻译生成的粗略草稿，质量较低。术语依照术语表，但句子可能有误；之后会由仔细的译文取代。

致读者

1. This series of volumes, a list of which is given on pages vii and viii, takes up mathematics at the beginning, and gives complete proofs. In principle, it requires no particular knowledge of mathematics on the reader’s part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course.

2. The method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular. This choice has been dictated by the main purpose of the treatise, which is to provide a solid foundation for the whole body of modern mathematics. For this it is indispensable to become familiar with a rather large number of very general ideas and principles. Moreover, the demands of proof impose a rigorously fixed order on the subject matter. It follows that the utility of certain considerations will not be immediately apparent to the reader unless he has already a fairly extended knowledge of mathematics; otherwise he must have the patience to suspend judgment until the occasion arises.

3. In order to mitigate this disadvantage we have frequently inserted examples in the text which refer to facts the reader may already know but which have not yet been discussed in the series. Such examples are always placed between two asterisks: \* ... \*. Most readers will undoubtedly find that these examples will help them to understand the text, and will prefer not to leave them out, even at a first reading. Their omission would of course have no disadvantage, from a purely logical point of view.

4. This series is divided into volumes (here called “Books”). The first six Books are numbered and, in general, every statement in the text

v 假定仅已知前几卷中已讨论过的结果。这个规则在每本书中都适用，但为了说明方便，这些书不再排列为连续的序。在每一本书（或这些章）的开头，读者都会找到其与其他书的逻辑关系的精确指示，因此他将能够满足自己不存在任何恶性循环。

5. The logical framework of each chapter consists of the definitions, the axioms, and the theorems of the chapter. These are the parts that have mainly to be borne in mind for subsequent use. Less important results and those which can easily be deduced from the theorems are labelled as “propositions”, “lemmas”, “corollaries”, “remarks”, etc. Those which may be omitted at a first reading are printed in small type. A commentary on a particularly important theorem appears occasionally under the name of “scholium”.

为了避免繁琐的重复，有时方便地引入记号或仅在章或章的某个截面内有效的缩写（对于例，在章中，仅与交换环，字相关） “环”始终表示“交换环”）。这些约定总是被明确提及，通常是在它们出现的章的开头。

6. Some passages in the text are designed to forewarn the reader against serious errors. These passages are signposted in the margin with the sign

☡

（“拐弯处危险”）。

7. The Exercises are designed both to enable the reader to satisfy himself that he has digested the text and to bring to his notice results which have no place in the text but which are nonetheless of interest. The most difficult exercises bear the sign ¶.

8. In general, we have adhered to the commonly accepted terminology, except where there appeared to be good reasons for deviating from it.

9. We have made a particular effort always to use rigorously correct language, without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses of language, without which any mathematical text runs the risk of pedantry, not to say unreadability.

10. Since in principle the text consists of the dogmatic exposition of a theory, it contains in general no references to the literature. Bibliographical references are gathered together in Historical Notes, usually at the end of each chapter. These notes also contain indications, where appropriate, of the unsolved problems of the theory.

VI

每个历史注记包含后面的参考书目一般只有那些在所讨论的理论的演变中具有最大的重要性的书籍和原始回忆录。它并没有假装完整；特别是，仅涉及确定优先问题的参考文献几乎总是被省略。 至于习题，我们认为总体上不值得指出它们的起源，因为它们来自许多不同的来源（原始论文、教科书、习题的收藏）。

11. 对这个级数的部分的引用如下： a) 如果引用*在同一截面*中出现的定理、公理或定义，则由其数引用。 b) 如果它们出现在*同一个章*的另一个截面中，则该截面也会在参考文献中引用。 c) 如果它们出现在*同一本书的另一个章中*，则引用章和截面。 d) 如果它们出现在*另一个书*中，则首先引用该书的书名。

结果摘要由字母 R 引用：因此 集合理论，R 表示“集合论 结果摘要”。

内容

的

数学的元素级数

I。 集合论

1. Description of formal mathematics. 2. Theory of sets. 3. Ordered sets; cardinals; natural numbers. 4. Structures.

II。 代数

1. 代数结构。 2. 线性代数。 3. 张量代数，外部代数，对称代数。 4.多项式和有理分式。 5.域。 6. 订购群和域。 7. 模超过主要理想环。 8.半单模和环。 9. 倍半线性和二次形式。

VII