The Basic Library List Committee considers this book essential for undergraduate mathematics libraries. Atiyah-MacDonald, as everyone in the world calls this classic book, is one of the premier texts for a serious graduate or very gifted undergraduate student aspiring to learn both commutative algebra for his PhD qualifying examination and much of the machinery required for algebraic geometry. These two statements require some elaboration: first, commutative algebra, as a mainstay of the algebra part of a solid PhD program, is treated wonderfully well in a number of texts, and many instructors would opt for other sources. For one thing, the chapters in Atiyah-MacDonald are cut to the bone: although the discussions overflow with elegance and abound with precision, they are not easy to use as scripts for accessible lectures. The lecturer would have to do a lot of work in order properly to motivate his presentations, at least for an average class.

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Papaioannou A. Introduction to Commut Commutative Algebra Pr Lecture 04 - Linear Article Submitted to J Combinatorics, Commuta Introduction to Linear The non-commutative We SOL Algebra Commut Orderings and Algorith We can now easily deduce that the sum of a nilpotent element and a unit is a unit itself. We deduce that a0 is a unit. We use induction on n to prove that the coe? If the proposition holds for n?

Hence an is nilpotent and we may apply the inductive hypothesis. The converse follows from exercise 1 and exercise 2, ii. Now the inductive hypothesis applies. Conversely, if all the coe? The converse is trivial. R, since R is the intersection of all prime ideals, while J is the intersection of all prime and maximal ideals.

Therefore, we only need to show J? R in A[x]. In particular 1? This completes the proof. Therefore, f? But the constant term of g x is a1, which must be nilpotent, too. By this process, we show that all the coe? The inverse is not true; a su? Conversely, given any ideal b of A[[x]], bc consists of all coe?

Let p be a prime ideal in A and let x be a non-zero element of A? This implies that a0x a0x? This implies that xn? Conversely, if a is an intersection of prime ideals, then this intersection is contained in the intersection of prime ideals of r a , hence r a?

But this implies that every prime ideal coincides with R since R is maximal and so there is only one prime ideal in A. The direction x, y? Construction of an algebraic closure of a? Artin 1. Assume the contrary. Then, m? This furnishes a contradiction to the maximality of m. These results show that the space Spec A of all prime ideals of A can be endowed with a topology - the Zariski topology - if we de? Therefore, there is a? Conversely, if an open subspace Y of X is a union of a?

By vii , each of those will have a? E would generate the unit ideal. The latter condition is ful? Therefore, X is irreducible if and only if the nilradical is prime. In a Hausdor? Y is dense in X?? Given a ring A, we have the following: i? This decomposition implies that X is disconnected. X be both open and closed. Since Y is open, it is a union of sets Xf. Since Y is a closed subspace of a quasi-compact space, it is quasi-compact too hence it is a?

Now, ii? Since fx is continuous, there is a neighborhood Ux of x on which fx does not vanish. By the compactness of X, a? We obviously have? Since the open sets Uf resp.

Uf form bases of the topologies on X and X we deduce that? Therefore X is homeomorphic to Max C X. Chapter 2 Modules 2. Therefore, given x? A M and the cokernel of incl? The above proposition is true even if M is not a? This is clearly a bilinear homomorphism, which induces a linear homomorphism M? A M , as desired Lang, Algebra, The condition M? But Mk and Nk are vector? Let without loss of generality the former be true. Indeed, surjectivity is obvious and we may write any arbitrary generator of a?

Any two such generators will map to the same element in a xm is and only if their respective? Hence the natural map is an isomorphism, as desired. A m and thus it induces an isomorphism of tensor products between A[x]? Now let p be a prime ideal of A. A N , since the tensor functor is associative tensoring an exact sequence?

By proposition 2. A f is injective, then so is the homomorphism f. For assume that 1? A f x2 hence y? A x2 by the injectivity of 1? But then, 1? Now if 0? The above lemma implies that we may remove the tensor by the? Hence N is? We see that if x1, x2,. But by the exactness of the sequence, Coker g M , hence M will be? A An, of course as a vector space over the? But, if x1, x2,. Choose u1, u2,. Since there is a?

In particular, it will be? Then, 1? Then, let p : NB? Direct limits 2. Then the modules Mi and homomorphisms? We shall construct an A-module M called the direct limit of the direct system M. Let C be the direct sum of the Mi; identify each module Mi with its embedding in C. The module M together with the family of homomorphisms?

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