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Posted By: pradha s Member Level: Gold Posted Date: 27 Jun 2008
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2007 Alagappa University M.Phil(Mathematics) Question paper
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DISTANCE EDUCATION
M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2007.
COMMUTATIVE ALGEBRA
(Upto 2006 Batch)
Time : Three hours Maximum : 100 marks
Answer any FIVE questions.
Each question carries 20 marks.
1. (a) Show that an R-module M is cyclic if and only if for same ideal I in R.
(b) Define a project module. Show that an R-module P is projective if and only if for any surjective homomorphism the induced homomorphism is surjective.
2. (a) Show that the tensor product of two free modules of rank m and n is free of rank .
(b) Give three definitions for faithfully flat module and show that they are equivalent.
3. (a) Show that is a unit for all .
(b) State and prove the Nakayama Lemma.
4. (a) Define and . Show that the defined by is a well defined isomorphism of -module.
(b) State and prove Hilbert Basis Theorem.
5. (a) Show that any finitely generated module over an Artinian ring is Artinian.
(b) State and prove the Jordan Holder Theorem.
6. (a) Show that the set of elements of S integral over R is a subring of S containing R.
(b) State and prove Noether's Normalisation theorem.
7. Let R be a Noetherian local domain with unique maximal ideal and K the quotient field of R. Show that the following are equivalent
(a) is a discrete valuation ring
(b) R is a principal ideal domain
(c) m is principal
(d) R is integrally closed and every non-zero prime ideal of R is maximal
(e) Every non-zero ideal of R is a power of m.
8. (a) Let be a homomorphism of R-modules, M and projective resolutions of respectively show that there exists a mapping of complexes such that moreover F is unique upto homotophy.
(b) Show that any R-module M can be embedded in an injective R-module.
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