Books on Noncommutative Algebra

In summary, there are several highly recommended books on noncommutative algebra, including Introduction to Noncommutative Algebra by Matej Bresar, Noncommutative Algebra by Benson Farb and R. Keith Dennis, A First Course in Noncommutative Rings by T.Y. Lam, and Introductory Lectures on Rings and Modules by John A. Beachy. Another great resource is Peter Cameron's "Introduction to Algebra," which covers a wide range of topics and has a specific section on noncommutative rings and algebras. These books are suitable for both advanced undergraduates and beginning graduate students.
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I would be interested to know from MHB readers, of any books on Noncommutative Algebra you have a high opinion of ... ... especially those that are at the advanced undergraduate and beginning graduate level ...The books I have on this topic are as follows:

Introduction to Noncommutative Algebra by Matej Bresar (Springer 2014)

Bresar states that "the purpose ... is to give a gentle introduction to noncommutative rings and algebras that requires fewer prerequisites than most other books on the subject. ... ... The necessary background to read this book is a standard knowledge of linear algebra and a basic knowledge about groups rings and fields. ... "Noncommutative Algebra by Benson Farb and R. Keith Dennis (Springer-Verlag 1993) [aimed at beginning gradate students]


A First Course in Noncommutative Rings by T.Y. Lam [aimed at students who have done a beginning graduate course in abstract algebra]Introductory Lectures on Rings and Modules by John A. Beachy [focussed on noncommutative aspects of rings and modules, aimed at advanced undergrads/beginning graduates]Basically, I would be interested in readers' opinions of the above books ... ... and I would be VERY interested to hear of any other noncommutative algebra books you have a high opinion of ...

Peter
 
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Cameron's "Introduction to Algebra" (Oxford University Press, 2008) is a great book for an introduction to noncommutative algebra. It covers a wide range of topics, including group theory, rings, modules, and fields, and has a section specifically on noncommutative rings and algebras. The writing style is clear and easy to understand, making it a great resource for both advanced undergraduates and beginning graduate students. I highly recommend it as a comprehensive introduction to noncommutative algebra.
 

1. What is noncommutative algebra?

Noncommutative algebra is a branch of mathematics that deals with structures where the order of multiplication matters. In contrast to commutative algebra, where the order of multiplication does not affect the result, noncommutative algebra studies objects such as matrices and operators that do not follow the commutative property.

2. What are some applications of noncommutative algebra?

Noncommutative algebra has many applications in various fields such as physics, engineering, and computer science. For example, it is used in quantum mechanics to describe the behavior of particles, in control theory to design efficient systems, and in coding theory to construct error-correcting codes.

3. What are the main concepts in noncommutative algebra?

The main concepts in noncommutative algebra include rings, modules, and algebras. These structures are studied in relation to their properties, such as ideals, homomorphisms, and representations. Other important concepts include tensor products, universal properties, and homological algebra.

4. What are some recommended books on noncommutative algebra?

Some recommended books on noncommutative algebra include "Noncommutative Algebra" by Benson Farb and R. Keith Dennis, "Introduction to Noncommutative Algebra" by Matej Bresar, and "Noncommutative Algebra for Beginners" by Jonathan K. Hodge and Steven G. Zucker. These books cover the fundamentals of noncommutative algebra and provide a comprehensive introduction to the subject.

5. What are some prerequisites for studying noncommutative algebra?

To study noncommutative algebra, a strong foundation in abstract algebra, including group theory, ring theory, and field theory, is necessary. Familiarity with linear algebra and basic concepts in topology and category theory may also be helpful. Additionally, a good understanding of mathematical proof techniques is essential for studying noncommutative algebra.

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