Model Question Paper www.alagappauniversity.ac.in Distance Education Alagappa University : MPhil Mathematics

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Document Described : Alagappa University
DISTANCE EDUCATION
M.Phil. (Mathematics) DEGREE EXAMINATION,
MAY 2011.
COMMUTATIVE ALGEBRA
(upto 2006 Batch)
Time : Three hours Maximum : 100 marks

http://www.alagappauniversity.ac.in/...9&college_id=2
http://www.alagappauniversity.ac.in/files/question_paper/2-M.Phil%20(Maths).doc

Answer any FIVE questions.
All questions carry equal marks.
(5 ? 20 = 100)

1.(a)If is a split exact sequence prove that .
(b)Prove that if is a projective module, then there exists a free module such that is free.

2.(a)Show that the tensor product of two modules exists and it is unique upto isomorphism.
(b) Let and be free modules with bases and , respectively. Then show that is free with basis .

3.(a)Show that is faithfully flat if and only if is flat and for each maximal ideal m of R, mM M.
(b)State and prove Chinese Remainder theorem.

4.(a)If is an Artinian ring prove that the nil radical N(R) is nilpotent.
(b)Show that Rs is flate R-module.

5.(a)Show that the nil radical of R extends to the nil radical of Rs.
(b)State and prove Hilbert’s basis theorem.

6.(a)State and prove the structure theorem for artinian rings.
(b)State and prove Jordan Hölder theorem.

7.(a)Prove that the ring of integers in an algebraic number field is Dedekind domain.
(b)Let R be a finitely generated k-algebra, k filed, and m a maximal ideal of R. Then prove that R/m is a finite extension of k.

8.(a)Let R be a Noetherian domain in which every non-zero prime ideal is maximal. Prove that the following statements are equivalent.
(i)R is Dedekind domain;
(ii)Rp is discrete valuation ring for every non-zero prime ideal, P of R;
(iii)Every primary ideal of R is a power of a prime ideal.
(b)Show that any R-module M can be embedded in an injective R-module.