Algebra (AMS Chelsea Publishing) by Saunders Mac Lane (PDF)

Description

This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance.

This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach–emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s–was popularized for the graduate level in the 1940s and 1950s to some degree by the authors’ publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.

User’s Reviews

Editorial Reviews: Review “Nearly every ten years there seems to arrive a new edition of this now classical book the review of which the reviewer hardly can improve. The main advantage of the authors had been the introduction of thoroughly categorical concepts into algebra.” —- Zentralblatt MATH”The book is clearly written, beautifully organized, and has an excellent and wide-ranging supply of exercises … contains ample material for a full-year course on modern algebra at the undergraduate level.” —- Mathematical Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐MacLane was, according to wikipedia, one of the two founders of category theory in 1942-1945. This 1961 algebra book by MacLane and Birkhoff applies the category theory perspective throughout, for better or worse. I’ll be marked down for saying this, but in my opinion, category theory is not necessarily a good idea. It is now the orthodox way to do algebra, and no university will let you do advanced algebra without it nowadays, although in the 1970s it was not universally taught as the standard framework. I didn’t encounter category theory until my early postgraduate years, and that was only very specifically in the context of algebraic topology. But if you want to be in the modern mainstream, you need the category theory perspective, and this book is probably the best way to get it.Now there’s a little oddity about what is really meant by “category theory”. Someone informed me a couple of months ago that I had made a grave error in my own definition of ring epimorphisms. Apparently the really modern definition of an epimorphism in category theory requires the right cancellative property. (And monomorphisms require the left cancellative property. See wikipedia for details.) But this book by MacLane and Birkhoff (page 38) uses the old-fashioned definition, which only requires surjectivity. In the case of the ring category, it makes a big difference. (The old-fashioned epimorphism definition is also given by Seth Warner “Modern algebra”, page 82; Lang “Algebra”, page 120; Grove “Algebra”, pages 8, 49, 127; Ash “Basic abstract algebra”, pages 19, 37, 89.) So if you want to use MacLane and Birkhoff for category theory, beware of this difference.It seems to me as a non-algebraist that category theory was introduced as a kind of “grand unified theory” to implement bulk handling of algebraic categories. But in my opinion, it is the differences between the categories which make them interesting, and when category theory is applied to locally Cartesian spaces, C^k differentiable manifolds and fiber bundles, for example, the effort required to fit the categories into the general framework of morphisms and functors is alien to the nature of the objects under study.In conclusion, if you don’t have this book, you’re not an algebraist. And if you’re some other kind of mathematician, it is the best all-round reference for all of the basic categories of modern abstract algebra. This book has pretty much everything a non-algebraist would want to know about the broad sweep of algebra. It even has chapters on multilinear and exterior algebra and affine spaces. It’s a tidy, carefully-written reader-friendly book which tries to help the reader, not show off the superior genius of the authors as some frustrating books do.By the way, as I understand it, MacLane only introduced the space between Mac and Lane because his wife couldn’t type a capital letter in the middle of a word. I think that’s a silly reason to change the spelling of one’s name. So I spell it in the traditional way as “MacLane”. Honi soit qui mal y pense!

⭐Even the expert will enjoy this book. Every page is packed with rich ideas explained in a simple, clear, concise way. If one is already familiar with the subject, one can read this book like a novel and take joy in revisiting known concepts or learning them in a new way. For the novice, this book builds algebra one atom at a time from the first principle.This book also gave me some insight into how MacLane (co-inventor) came to categorical ideas. There is nothing like having the master show you in his words (and much credit to the co-author Birkhoff) the beautiful landscape and richness of algebra.

⭐This algebra book is classic and stunning. It is a small fraction the cost of modern algebra books but the 3rd edition is near perfection. My only nit is I prefer the ‘blackboard’ notation versus the bold typeface for vectors and fields. It takes me about two hours to read each page as the information context is so rich. I have to retrace and Google something about every 8-10 lines. It is going to take me about six months of full time study to read this book.

⭐Herewith, the dumbest (preliminary, in the sense of I just got it in the mail today) review of MacLane and Birkhoff, 3rd Edition that you’ll read. This book is very well made, with quality paper, and a buckram binding Smyth-sewn in signatures. Also from a more substantive POV: The exposition seems clear. More to come as I work through the text. I guess some reviewers have read all 600 pages and worked all the problems? Fibbers!

⭐Really a high quality book.The condition of the book was as close to perfect as i could tell

⭐There is so much heart poured in this book, just beautifully written and intended for all things good. I ended up buying a second copy just to have a new one.

⭐Arrived early and as advertised.

⭐i used this text as my first algebra book. if you just began math and want to see a solid introduction to modern math this is it; self-contained and introduces many important ideas that are commonly used in modern math but typically not taught in undergraduate math courses.

⭐I got a photo copy of the first 3 chapters of this book as part of my B.Sc in Mathematics. Having the the complete book is a delight. The printing quality and binding are sublime. The covering of the subjects are masterfully explained with a moderate amount of exercises. Some math books are spammed with exercises. Here you get just enough to get your understanding going. Its a good reference when you want to look up any basic algebra concepts. I don’t think this book will ever get outdated, but maybe if computers take over more and more deductive work from mathematicians an appendix covering this could be added.

⭐Uma obra-prima! Um livro que não pode faltar na biblioteca de um estudioso da Matemática.Ce livre est d’une qualité exceptionnelle pour qui veut aborder l’algèbre. L’exposé est progressif et clair. Il part des notions élémentaires au chapitre I, des groupes et des anneaux aux chapitres II et III respectivement. Le chapitre IV est inhabituel et fait toute l’originalité du livre: bien que celui-ci soit destiné à des débutants (disons, de niveau L1 ou L2), les auteurs introduisent les “constructions universelles”, en l’occurrence des notions élémentaires sur les catégories. Elles seront complétées au chapitre XV par la théorie des foncteurs. C’est muni de ce langage qu’aux chapitres V à IX les modules, les espaces vectoriels, les matrices, les déterminants et les produits tensoriels sont traités (attention: la matrice de passage est l’inverse de celle que l’on définit habituellement). Les transformations naturelles comme l’isomorphisme canonique entre un espace vectoriel de dimension finie et son bidual, les objets universels comme les produits tensoriels, peuvent donc être présentés dans ce langage, qui est tellement plus parlant que celui de Bourbaki! Viennent ensuite l’étude des formes bilinéaires et quadratiques au chapitre X, et au chapitre XI la théorie des modules de type fini sur les anneaux principaux commutatifs. C’est à cette occasion qu’est traitée la forme de Jordan, et je trouve qu’en effet ce résultat essentiel n’est clair qu’à la lumière de la théorie des modules. La structure des groupes est traitée au chapitre XII, permettant ainsi l’étude de la théorie de Galois au chapitre XIII. L’exposé de la théorie de Galois me paraît être une des meilleures introductions à ce beau sujet. Il se termine par le théorème de Ruffini-Abel (impossibilité de résoudre par radicaux les équations algébriques de degré supérieur à 4). Le chapitre XIV sur les treillis (avec notamment le théorème de Jordan-Hölder-Dedekind) est lui aussi excellent. Le chapitre XVI et dernier du livre est consacré à l’algèbre multilinéaire, en particulier l’algèbre extérieure, indispensable pour tous ceux qui veulent étudier le calcul tensoriel. L’appendice sur les espaces affines et projectifs ne me paraît pas inutile (surtout pour une prise de contact avec les espaces projectifs, omniprésents en Géométrie algébrique).I bought this book cos I though I could learn much more about math with this book and indeed I was right. I learnt a lot of new stuff which I’ve never seen even in the secondary school. It’s worth buying the book cos adds more knowledge to the subject and explain the things quite clearly. If you want to learn more stuff or remember what you learnt in the algebra lessons from you secondary school, this is the correct book for you. It’s worth buying the book if you really like mathematics and algebra.

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