By Ernst Kunz (auth.)

ISBN-10: 1461459869

ISBN-13: 9781461459866

ISBN-10: 1461459877

ISBN-13: 9781461459873

Originally released in 1985, this vintage textbook is an English translation of *Einführung in die kommutative Algebra und algebraische Geometrie*. As a part of the fashionable Birkhäuser Classics sequence, the writer is proud to make *Introduction to Commutative Algebra and Algebraic Geometry* on hand to a much broader audience.

Aimed at scholars who've taken a uncomplicated direction in algebra, the objective of the textual content is to give very important effects about the illustration of algebraic forms as intersections of the least attainable variety of hypersurfaces and—a heavily similar problem—with the main low-cost new release of beliefs in Noetherian earrings. alongside the best way, one encounters many simple innovations of commutative algebra and algebraic geometry and proves many proof which may then function a uncomplicated inventory for a deeper research of those subjects.

**Read Online or Download Introduction to Commutative Algebra and Algebraic Geometry PDF**

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**Extra resources for Introduction to Commutative Algebra and Algebraic Geometry**

**Example text**

It is a prime ideal of R. Proof. If {J~heA is a totally ordered (with respect to inclusion) family of ideals of M, then J := U~eA J~ is an ideal of R with I C J and J n S = 0. By Zorn's Lemma M has a maximal element p. Suppose that for two elements at, a 2 E R \ p we have at · a 2 E p. Since a, f/. p, we have (Ra, + p) n S :F 0 (i = 1, 2). Hence there are elements r, E R, Pi E P such that (i = 1,2). But then (rtat + pt) · (r2a2 + P2) = rtr2a1a2 + r1a1P2 + r2a2P1 + PtP2 E P n S, contradicting p n S = 0.

Proof. The statement about the kernel follows immediately from the definition of tp and the definition of the direct product of rings. We prove the surjectivity of tp by induction on n. Let n = 2 and (rt +It, r2 + /2) E R/ It x R/ /2 be given. We have an equation 1 =at + a2 with ak E Ik 1 mod lz ( l "# k). If we put r := r2at + rta2, then (k = 1, 2) and so ak r rk mod Ik (k = 1, 2) which proves that tp is surjective for n = 2. Now let n > 2 and suppose the proposition has been proved for fewer than n pairwise relatively prime ideals.

1} be pairwise relatively prime ideals of a ring R. Then the canonical ring homomorphism cp: R-+ R/h X .. 7. (Chinese Remainder Theorem) Let /1, ... , In (n r~--+ (r+h, ... ,r+ln) 42 CHAPTER II. DIMENSION is an epimorphism with kernel n~=t /t. Proof. The statement about the kernel follows immediately from the definition of tp and the definition of the direct product of rings. We prove the surjectivity of tp by induction on n. Let n = 2 and (rt +It, r2 + /2) E R/ It x R/ /2 be given. We have an equation 1 =at + a2 with ak E Ik 1 mod lz ( l "# k).

### Introduction to Commutative Algebra and Algebraic Geometry by Ernst Kunz (auth.)

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