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During these days, I have been reading some books concerning algebraic geometry. But I learned commutative algebra two years ago, I cannot remember some basic things such as the definition of irreducible ideals.

Can anyone recommend me a rigorous book to help me recover those old knowledge? I hope it's a book in GTM series.

Thanks.


Thanks for yours answers!! I will read those books!!

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marked as duplicate by rschwieb abstract-algebra Jul 29 at 19:32

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ The following link may help you, although I know this is not what you are asking for. math.stackexchange.com/questions/255063/… $\endgroup$ – NoChance Jul 29 at 3:00
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    $\begingroup$ I was only able to add five duplicates, although I had a lot more queued up. Just putting "[book-recommendation] gallian" into the search field brought a bunch to the top. The references within should be more than sufficient. And if not those, try the boatload of duplicates I wasn't able to list... just use the search bar. $\endgroup$ – rschwieb Jul 29 at 19:35
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In addition to Joseph Gallian's, Contemporary Abstract Algebra, I would also like to recommend John B. Fraleigh's, A First Course in Abstract Algebra for the more advanced student.

Although it is indeed, an undergraduate text, I found it to be a nice bridge between undergraduate and graduate abstract algebra, as I often consulted it for good, basic, explanations of concepts that were not presented clearly in my graduate text.

One of my favorite anecdotes is Fraleigh's take on the number $\mathbf{1}$: ``Unfortunately, $i$ has been called an imaginary number and this terminology has led generations of students to view complex numbers with more skepticism than the real numbers. Actually, $all$ numbers, such as $1$, $3$, $\pi$, $-\sqrt{3}$, and $i$ are inventions of our minds. There is no physical entity that is the number $1$. If there were, it would surely be in a place of honor in some great scientific museum, and past it would file a steady stream of mathematicians, gazing at $1$ in wonder and awe.''

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You can follow this one

"Contemporary Abstract Algebra" by Joseph A. Gallian

It's an excellent book for a strong base for algebra group theory, ring theory, field theory, vector spaces, eigen values eigen vectors, canonical forms. The great thing is that this book talks to you. It is interactive. Chapter are small in size and there are a lot of examples for each concept. Once you go through them you will never forget the theory.

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If you’re just looking to re-up your commutative algebra, you may want to consider these notes by Gathmann.

If you’re looking for a general algebra book, I’d recommend Algebra: Chapter 0 by Aluffi. It gives a categorical perspective on algebra, which may be useful if you’re looking to move from algebra to algebraic geometry. It’s also just a great book.

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