By Gregor Kemper
This textbook deals a radical, smooth creation into commutative algebra. it truly is intented ordinarily to function a advisor for a process one or semesters, or for self-study. The rigorously chosen material concentrates at the options and effects on the middle of the sphere. The publication keeps a relentless view at the typical geometric context, allowing the reader to achieve a deeper figuring out of the fabric. even though it emphasizes idea, 3 chapters are dedicated to computational features. Many illustrative examples and workouts enhance the text.
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Extra resources for A Course in Commutative Algebra
A morphism f : X → Y is called an isomorphism if there exists a morphism g: Y → X with f ◦ g = idY and g ◦ f = idX . , a topological isomorphism). 21 (a)). Let f : X → Y be a morphism given by polynomials f1 , . . , fn . Then we have a homomorphism of K-algebras ϕ: K[Y ] → K[X] given as follows: If K[X] = K[x1 , . . , xm ]/I(X) and K[Y ] = K[y1 , . . , yn ]/I(Y ), then ϕ (yi + I(Y )) := fi + I(X). It is routine to check that this is well deﬁned. The homomorphism ϕ is said to be induced from f .
To get the proof for the case of Artinian modules, replace every occurrence of the word “ascending” in the above argument by “descending,” and exchange “Mi ” and “Mn ” in the proof of Mi = Mn . We need the following deﬁnition to push the theory further. 5 (Ideal product). Let R be a ring, I ⊆ R and ideal, and M an R-module. (a) The product of I and M is deﬁned to be the abelian group generated by all products a · m of elements from I and elements from M . So n ai mi n ∈ N, ai ∈ I, and mi ∈ M . IM = i=1 Clearly IM ⊆ M is a submodule.
B) R is Noetherian and every prime ideal of R is maximal. 8 can be rephrased as, “R is Noetherian and has dimension 0 or −1” (where −1 occurs if and only if R is the zero ring). We prove only the implication “(a) ⇒ (b)” here and postpone the proof of the converse to the end of Chapter 3 (see page 42). Proof of “ (a) ⇒ (b)”. Suppose that R is Artinian. The ﬁrst claim is that R has only ﬁnitely many maximal ideals. Assume the contrary. Then there exist inﬁnitely many pairwise distinct maximal ideals m1 , m2 , m3 , .
A Course in Commutative Algebra by Gregor Kemper