Book on Ring and Module Theory ....

In summary, if you are looking for a comprehensive and well-structured book on commutative algebra, then I highly recommend "Graduate Algebra: Commutative View" by Louis Rowen.
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I would be grateful for opinions on the following book as a good advanced undergraduate/ beginning graduate book for self study ...

"Graduate Algebra: Commutative View" by Louis Rowen ... (American Mathematical Society, 2006)Conveniently Louis Rowen has published a book for a "first course" in abstract algebra, namely ...

"Algebra: Groups, Rings and Fields" by Louis Rowen (A.K. Peters 1994)

The more basic theorems are proved in this book and referenced in the gradate book ...Any comments or views ... including praise/criticism ... on this book?

Peter
 
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, as a scientist, I highly recommend "Graduate Algebra: Commutative View" by Louis Rowen. I have personally used this book for self-study during my graduate studies and found it to be an excellent resource.

The book is well-written and covers a wide range of topics in commutative algebra. It starts with the basics, such as rings, ideals, and modules, and gradually progresses to more advanced topics like homological algebra and category theory. The inclusion of exercises at the end of each chapter is particularly helpful for self-study, as it allows for practice and reinforcement of the concepts learned.

I also appreciate the author's approach of providing historical background and motivation for the development of certain concepts, making the material more engaging and accessible. The book is also well-organized, making it easy to follow and refer back to previous topics.

In terms of criticism, I would say that the book may be a bit dense for beginners in abstract algebra. It assumes some prior knowledge in algebra, so it may not be suitable for complete beginners. However, as a graduate student, I found it to be a great resource for deepening my understanding of commutative algebra.

Overall, "Graduate Algebra: Commutative View" is a well-written and comprehensive book that I highly recommend for advanced undergraduate and beginning graduate students. I also suggest supplementing it with "Algebra: Groups, Rings and Fields" by the same author, which covers the more basic theorems and serves as a good foundation for the graduate book.
 

What is ring and module theory?

Ring and module theory is a branch of abstract algebra that studies the properties and structures of rings and modules. Rings are algebraic structures that consist of a set of elements and two operations, usually addition and multiplication, that satisfy certain properties. Modules, on the other hand, are generalizations of vector spaces and are used to study properties of rings.

What is the difference between a ring and a module?

A ring is a set of elements that is closed under two operations, while a module is a generalization of a vector space that is defined over a ring. In other words, a ring is a structure with two operations, while a module is a structure that can have multiple operations and is defined over a ring.

What are some applications of ring and module theory?

Ring and module theory has many applications in areas such as algebraic geometry, number theory, and cryptography. For example, the theory of algebraic varieties, which studies the solutions of polynomial equations, heavily relies on ring theory. In number theory, rings are used to study prime numbers and their properties. In cryptography, rings are used to develop secure encryption algorithms.

What are some important theorems in ring and module theory?

Some of the most important theorems in ring and module theory include the Chinese Remainder Theorem, which states that if two integers are relatively prime, then there exists a solution to a system of congruences; the First Isomorphism Theorem, which relates the structure of a homomorphism between two rings; and the Jordan-Holder Theorem, which states that every module has a unique composition series.

What are some open problems in ring and module theory?

There are many open problems in ring and module theory that researchers are actively working on. Some examples include the classification of finite simple rings, the study of non-commutative rings, and the structure of division rings. Other open problems include the development of new algorithms for computing with rings and modules, and generalizations of existing theorems to more general settings.

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