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Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson´s structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients.

Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.

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Finite Dimensional Division Algebras.- Structure of Finite Dimensional Algebras.- Modules and Vector Spaces.- Tensor Products.- Structure of Rings.- Noncommutative Polynomials.- Rings of Quotients and Structure of PI-Rings.

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"The abundant and carefully chosen examples in the text are selected in such a way as to illustrate the exposition and give some insights into what is going on. ... The book is very good. It will soon find its place in classrooms for most courses in ring theory. I personally liked it very much, and in 2014 our department included the book in the principal bibliography for the corresponding graduate course in noncommutative algebra." (Plamen Koshlukov, Mathematical Reviews, 2018)

"´Introduction to noncommutative algebra´ is a very well written book and it is very pleasant to read. ... I was very much impressed by the lists of exercises given at the end of each chapter as some of the exercises are not found in standard texts on ring theory." (Veereshwar A. Hiremath, zbMATH 1334.16001, 2016)

"Students of mathematics typically encounter examples of noncommutative algebraic structures in a first course in abstract algebra, but systematic study of such objects is reserved for a more advanced course. This offering by Bresar (Univ. of Ljubljana, Slovenia) is written in a style governed by two guiding principles. ... an excellent choice for a first graduate course in noncommutative algebra. Summing Up: Highly recommended. Graduate students and researchers/faculty." (D. S. Larson, Choice, Vol. 52 (10), June, 2015)

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Matej Bresar is Professor of Mathematics at University of Ljubljana and University of Maribor. He is the author or co-author of over 140 research papers whose topics are mostly on noncommutative algebra and its applications. He is also the co-author of the book Functional identities (Birkhauser, 2007).

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