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I've been studying from the book Noncommutative Rings by Herstein (not as a part of some official course), but unfortunately it doesn't contain any exercises apart from a few simple ones in the body. I was wondering if anyone had any suggestions on where I could find suitable exercises (since from what I've seen, this material is covered quite differently in different books, so it may not necessarily be possible to just take any other book on noncommutative algebra and use its exercises).

Any help will be appreciated.

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Lam's books are great. A first course on non-commutative rings is a wonderful book that contains a wealth of examples and exercises. These are solved and discussed in detail in the complementary Exercises books (I don't remember the exact title). –  t.b. Apr 9 '11 at 13:59
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Exercises in classical Ring Theory is the companion to First Course. The companion to Modules and Rings (which in a very real sense is the Second Course in non-commutative rings) is "Exercises in Modules and Rings". Both are in the Springer "Exercises" series. –  Arturo Magidin Apr 9 '11 at 19:31

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Louis Halle Rowen's books "Ring Theory" (Volumes I and II) are excellent comprehensive introductions to the theory of noncommutative rings and bring the reader up to a level from which he could begin research in the subject. (These books are more comprehensive than Herstein.) Furthermore, there are a wealth of examples and exercises in these books which is really quite rare for mathematics texts at this advanced level.

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