Artin Algebra

[Rolando Kumar]

Michael Artin's "Algebra" is a textbook that covers various topics in abstract algebra, including group theory. It is commonly used as a reference for undergraduate and graduate level courses in mathematics.

Artin's "Algebra" provides a comprehensive and rigorous treatment of group theory, making it an excellent resource for learning the fundamentals of this branch of mathematics. It includes numerous examples, exercises, and applications to help students understand the concepts and develop problem-solving skills.

While "Algebra" is primarily used as a textbook for courses, it can also be used for self-study. However, it is recommended that the reader have a strong background in mathematics, particularly in linear algebra and abstract algebra, before attempting to study group theory from this book.

Artin's "Algebra" covers a wide range of topics in group theory, including subgroups, homomorphisms, normal subgroups, and factor groups. However, it may not be suitable for advanced topics such as representation theory or Galois theory. It is best to consult with a more specialized textbook for these topics.

Yes, there are various online resources and lecture notes available that can supplement the material covered in "Algebra" for group theory. Additionally, there are other textbooks that may provide a different perspective on the subject and can be used in conjunction with "Algebra" for a more comprehensive understanding of group theory.

The main differences between the latest edition of Artin's Algebra and the previous editions include updates to the content, new examples and exercises, and improvements in the organization and clarity of the material. The latest edition also includes new sections on topics such as group theory and Galois theory, which were not covered in previous editions.

Yes, there have been some changes in the notation used in the latest edition of Artin's Algebra. These changes were made to improve consistency and clarity in the presentation of concepts. Some of the changes include using a bold font for vectors, using a tilde for isomorphism, and using "iff" instead of "if and only if."

The latest edition of Artin's Algebra is suitable for both self-study and classroom use. While it is primarily designed as a textbook for undergraduate courses in abstract algebra, it is also a valuable resource for self-study and review. The book includes numerous examples and exercises, as well as a solutions manual for instructors.

The latest edition of Artin's Algebra has a slightly different format and layout compared to previous editions. The book now has a more streamlined and modern design, with color-coded sections and headings for easier navigation. Additionally, the latest edition includes more figures and diagrams to aid in understanding the material.

It is not necessary to purchase the latest edition if you already have a previous edition of Artin's Algebra. However, if you are using the book for a course or are interested in the new content and improvements, it may be beneficial to upgrade to the latest edition. Keep in mind that the core concepts and principles remain the same in all editions, so you can still use a previous edition for self-study or review.

Currently a 5th year PhD student, and I've been fighting tooth and nail to teach one of our junior-year honors sections in undergraduate algebra next fall (desperately hopeful we'll be able to return to normal in-person lectures at that point). Looks like it may come to fruition. Some questions on texts.

When I took undergraduate algebra, it was taught out of Herstein's classic Topics in Algebra for a two-semester honors sequence. The reception was somewhat lukewarm from the students. I definitely recall a point towards the end of the second semester (Galois theory) when about half the class was more-or-less completely lost, and from what I remember I was a bit lost myself but still managed to do well in the course. I liked the book at the time, but I remember feeling like it was times a bit slick, bit generally a good read overall and I learned a ton.

Recently, I've been made aware of another text by Dan Saracino titled Abstract Algebra: An Introduction, and apparently it's well-regarded by some people I respect. I've ordered a copy, but in the meantime I'm curious what others' opinions are on a comparison between the two, and experience with students' performance using both? In previous years, the course has used Herstein (and I believe at one point Artin, another great option but with a very different style).

The course is aimed at some pretty bright students, many of whom will pursue PhDs themselves, so I'm tempted to use Herstein (the problems are great, albeit at times very challenging, and it's one of the standards for this purpose). At the same time, I'm always of the opinion that students learn best when the exposition is clear and well-motivated, and it sounds like Saracino is a good candidate for that.

Any advice or anecdotes for people who have used both texts? I'm very passionate about teaching and I'd like to put the students in a good position for future graduate work. At the same time, I want the material to actually stick. I guess everyone has their own style, but I prefer courses that follow a text pretty closely rather than relying only on lecture notes.

For context, most students will be coming from a history of similar honors-type courses (including a recent course in Linear Algebra from Axler's excellent book, and an analysis course from baby Rudin, so they're well-versed in proof-writing and have a bit of maturity). It sounds like about half of students are expected to have taken a general intro to abstract algebra course (basic group theory, rings, and vector spaces) and half will be seeing it for the first time.

Thanks for the advice everyone. After looking at many options and talking with faculty, it's been decided that we're using Lang's Undergraduate Algebra. I was leaning heavily towards Artin the past few days, but I was actually pleasantly surprised when I flipped through Lang's Undergraduate Algebra... far more standard organization, great exercises, and exposition is clear. It wasn't even on my radar (I was only familiar with the graduate version...) but this was strongly suggested by my advisor and it looks really fantastic.

In 1904, Macaulay described the Hilbert function of the intersection of two plane curve branches: It is the sum of a sequence of functions of simple form. This monograph describes the structure of the tangent cone of the intersection underlying this symmetry. Iarrobino generalizes Macaulay's result beyond complete intersections in two variables to Gorenstein Artin algebras in an arbitrary number of variables. He shows that the tangent cone of a Gorenstein singularity contains a sequence of ideals whose successive quotients are reflexive modules. Applications are given to determining the multiplicity and orders of generators of Gorenstein ideals and to problems of deforming singular mapping germs. Also included are a survey of results concerning the Hilbert function of Gorenstein Artin algebras and an extensive bibliography.

Passing from commutative rings to their spectra (in the sense of algebraic geometry), Artinian local algebras correspond to infinitesimal pointed spaces. As such, they appear as bases of deformations in infinitesimal deformation theory. For instance Spec(?[ϵ]/(ϵ 2))Spec(\mathbbK[\epsilon]/(\epsilon^2)) is the base space for 1-dimensional first order deformations. Similarly, Spec(?[ϵ]/(ϵ n+1))Spec(\mathbbK[\epsilon]/(\epsilon^n+1)) is the base space for 1-dimensional nn-th order deformations.

An Artinian local algebra has a unique prime ideal, which means that its spectrum consists of a single point, i.e., Spec(A)Spec(A) is trivial as a topological space. It is however non-trivial as a ringed space, since its ring of functions is AA. By this reason spectra of Artinian local algebras are occasionally called fat points in the literature.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles.

Algebra, 2nd Edition, by Michael Artin, provides comprehensive coverage at the level of an honors-undergraduate or introductory-graduate course. The second edition of this classic text incorporates twenty years of feedback plus the author's own teaching experience. This book discusses concrete topics of algebra in greater detail than others, preparing readers for the more abstract concepts; linear algebra is tightly integrated throughout.

For his leadership in modern algebraic geometry, including three major bodies of work: tale cohomology; algebraic approximation of formal solutions of equations; and non-commutative algebraic geometry.

N2 - We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring [Formula: see text] as the kernels of certain functors [Formula: see text] rather than of functors [Formula: see text] which are given by a pp-pair. This paper will give various algebraic characterizations of these functors in the case that [Formula: see text] is an artin algebra. Suppose that [Formula: see text] is an artin algebra. An additive functor [Formula: see text] preserves inverse limits and [Formula: see text] is finitely presented if and only if there is a sequence of natural transformations [Formula: see text] for some [Formula: see text] which is exact when evaluated at any left [Formula: see text]-module. Any additive functor [Formula: see text] with one of these equivalent properties has a definable kernel, and every definable subcategory of [Formula: see text] can be obtained as the kernel of a family of such functors. In the final section, a generalized setting is introduced, so that our results apply to more categories than those of the form [Formula: see text] for an artin algebra [Formula: see text]. That is, our results are extended to those locally finitely presented [Formula: see text]-linear categories whose finitely presented objects form a dualizing variety, where [Formula: see text] is a commutative artinian ring.

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