# A Course in Commutative Algebra (Graduate Texts in by Gregor Kemper

By Gregor Kemper

This textbook deals a radical, glossy creation into commutative algebra. it truly is intented regularly to function a consultant for a process one or semesters, or for self-study. The conscientiously chosen subject material concentrates at the ideas and effects on the heart of the sector. The ebook keeps a continuing view at the average geometric context, permitting the reader to achieve a deeper figuring out of the cloth. even though it emphasizes idea, 3 chapters are dedicated to computational facets. Many illustrative examples and workouts enhance the textual content.

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With ϕ: M → M/N the canonical epimorphism, set Mi := ϕ−1 (Ui ). This yields an ascending chain of submodules of M . By hypothesis, there exists an n with Mi = Mn for i ≥ n. Since ϕ(Mi ) = Ui , it follows that Ui = Un for i ≥ n. So we have shown that (a) implies (b). Now assume that (b) is satisﬁed. To show (a), let M1 , M2 , . . ⊆ M be an ascending chain of submodules. We obtain an ascending chain ϕ(M1 ), ϕ(M2 ), . . ⊆ M/N of submodules of M/N . Moreover, the intersections N ∩ Mi ⊆ N yield an ascending chain of submodules of N .

Proof. Assume that M is Noetherian, but there exists S ⊆ M that does not satisfy (b). We deﬁne ﬁnite subsets Si ⊆ S (i = 1, 2, . ) recursively, starting with S1 = ∅. Suppose Si has been deﬁned. Since S does not satisfy (b), there exists mi+1 ∈ S \ (Si )R . Set Si+1 := Si ∪ {mi+1 }. ) By (Si+1 )R for all i, contradicting (a). So (a) construction we have (Si )R implies (b), and it is clear that (b) implies (c). So suppose that (c) holds, and let M1 , M2 , . . ⊆ M be an ascending chain of submodules.

8 (Artinian and Noetherian rings). Let R be a ring. Then the following statements are equivalent: (a) R is Artinian. (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.

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