What is the best textbook for learning Ring Theory?

[JasonRox]

Does there exist a solid concrete detailed textbook on Ring Theory?

I'm looking for something like Rotman's book of Group Theory.

My background consists of basic Ring Theory/Field Theory from Herstein's Abstract Algebra.

I have Rowen's book in my hands right now (library), but it's far too advanced and it skips a lot (that's the purpose of the textbook I know).

Anything like Rotman's book would be great because I find Rotman's book to include everything in a detailed fashion without conversing too much, and simply sticking to definitions, theorems, proofs, results and remarks and the occasional motivation at the beginning and end of a chapter. Also, I find the questions to be just right. Some textbooks just have so many problems it's ridiculous. Sure lots of problems is good, but it's not necessary and there exists problem books for this purpose. Rotman uses the right problems and the right number of them, and often refers to them later which is great. Otherwise some books have like 50+ questions at the end of each section and half of the time you don't do the questions the textbook later refers to. It's darn annoying. I think the key to a good textbook is also to have the right/useful problems. Rotman's textbook only has like 6-7 per section on average and I learned more out of that textbook than out of Gallian's and Herstein.

Also, Rotman doesn't focus too much on applications either, but he does include interesting ones but also does not make those sections a necessity for later (other textbooks sometimes do by posing lots of questions based on applications). For example, Rotman does included Burnside's Lemma, but not knowing how this Lemma can be applied to certain combinatorial problems will have absolutely no impact on your progress through the book.

Yes, I praise Rotman's textbook.

I'm currently getting this because it contains material on Artinian rings, simple rings and such.

https://www.amazon.com/dp/1852332069/?tag=pfamazon01-20

Where can I go from here?

My goal is to have a strong algebra background.

 

[pivoxa15]

Hungerford is usually good with problems of a range of difficulty. Also try Fraleigh's A first course in abstract algebra which is more advanced then Hungerford.

As an aside, do you find doing algebra boring at times?
It's good in that one can get a real sense of understanding due to it being discrete but it gets boring doing all these problem yourself?

 

[morphism]

Ring theory is a broad field (!). What kind of topics are you looking to cover?

When I took basic ring theory, I used Herstein (Topics in Algebra), Dummit & Foote and Hungerford. D&F was pretty good, but it does fall under the "too many exercises per section" category. Hungerford doesn't have as many problems (but definitely has more per section than Rotman does), and on average has more challenging ones, but the text itself is dry. I thought Herstein was good (it's my favorite algebra book) but not as extensive as the other two (e.g. no talk of Artinian and Noetherian rings, Groebner bases, Hilbert's basis theorem or Nullstellensatz, localization, and a few other things that were parts of the course). Next term I'm going to be taking a second course on rings, and these are some textbooks that were suggested to me:
Noncommutative Rings, Herstein
A First Course in Noncommutative Rings, T.Y. Lam
Introductory Lectures on Rings and Modules, Beachy

pivoxa: Which Hungerford book are you talking about? Hungerford's Algebra is much more advanced than Fraleigh.

 

[pivoxa15]

Beginner Hungerford.

 

[Chris Hillman]

Does there exist a solid concrete detailed textbook on Ring Theory?

I'm looking for something like Rotman's book of Group Theory.

My background consists of basic Ring Theory/Field Theory from Herstein's Abstract Algebra.

In addition to books already mentioned, you might try any of these textbooks at various levels:

P.M. Cohn, Introduction to Ring Theory, Springer undergraduate mathematics series, Springer, 2000

Donald S. Passman, A Course in Ring Theory, Wadsworth & Brooks, 1991.

Louis H. Rowen, Ring Theory, Academic Press, 1991

Behrens, Ernst-August Behrens, Ring Theory, Academic Press, 1972.

A problem book:

T. Y. Lam, Exercises in Classical Ring Theory, 2nd ed., Springer, 2003.

Older but probably worth a look:

I. N. Herstein, Topics in Ring Theory, University of Chicago Press, 1969.

Irving Kaplansky, Notes on Ring Theory, University of Chicago, 1965.

You can also skim monographs and conference proceedings, e.g. graded rings, differential rings, computational algebra topics related to rings are all important for various reasons. E.g. try

K. R. Goodearl, Ring theory: nonsingular rings and modules, M. Dekker, 1976.

C. Nastasescu and F. van Oystaeye, Graded ring theory, North-Holland, 1982.

If you know what a group ring is, try:

Gregory Karpilovsky, Unit Groups of Group Rings, Longman,1989.

If you are interested in algebra, it is never too early to start playing with GAP.

 

[JasonRox]

In addition to books already mentioned, you might try any of these textbooks at various levels:

P.M. Cohn, Introduction to Ring Theory, Springer undergraduate mathematics series, Springer, 2000

Donald S. Passman, A Course in Ring Theory, Wadsworth & Brooks, 1991.

Louis H. Rowen, Ring Theory, Academic Press, 1991

Behrens, Ernst-August Behrens, Ring Theory, Academic Press, 1972.

A problem book:

T. Y. Lam, Exercises in Classical Ring Theory, 2nd ed., Springer, 2003.

Older but probably worth a look:

I. N. Herstein, Topics in Ring Theory, University of Chicago Press, 1969.

Irving Kaplansky, Notes on Ring Theory, University of Chicago, 1965.

You can also skim monographs and conference proceedings, e.g. graded rings, differential rings, computational algebra topics related to rings are all important for various reasons. E.g. try

K. R. Goodearl, Ring theory: nonsingular rings and modules, M. Dekker, 1976.

C. Nastasescu and F. van Oystaeye, Graded ring theory, North-Holland, 1982.

If you know what a group ring is, try:

Gregory Karpilovsky, Unit Groups of Group Rings, Longman,1989.

If you are interested in algebra, it is never too early to start playing with GAP.

I already looked the bold books. I already said I'm buying the first one.

The one by Passman looks good. I saw it at the library today. I'm going to look at it after my first exam.

I already have GAP and did an RA job requiring it. Nice little problem to do quick research for anyone.

 

[zhentil]

It seems to me that the jump between Herstein and any book specializing in ring theory would be enormous. Herstein's book is for a beginning course in undergraduate algebra, and almost every book on ring theory is aimed at people who have already taken a graduate algebra course. I second the recommendation for D&F. If you want a solid background in algebra, there's no better place to get it. If that's a little too basic for you, try Hungerford's grad-level Algebra book.

 

[JasonRox]

It seems to me that the jump between Herstein and any book specializing in ring theory would be enormous. Herstein's book is for a beginning course in undergraduate algebra, and almost every book on ring theory is aimed at people who have already taken a graduate algebra course. I second the recommendation for D&F. If you want a solid background in algebra, there's no better place to get it. If that's a little too basic for you, try Hungerford's grad-level Algebra book.

I got Hungerford's and it's readable.

I talked to my prof. and he said Herstein isn't any good anymore because it's old and such.

 

[morphism]

I got Lam's book from the library the other day. So far it's pretty good.

 

[pivoxa15]

Have you tried Schaum's outline book on algebra?