Tamil Nadu Open University (TNOU) 2006 M.Phil Mathematics Commutative Rings - Question Paper
MPL-638 RMSV
M.Phil. DEGREE EXAMINATION - JUNE 2006. Mathematics COMMUTATIVE RINGS Time : 3 hours Maximum marks : 75
Answer any FIVE questions.
Each question carries 15 marks.
1. (a) Let A be a ring * 0. Show that the set of prime ideals of A has minimal elements with respect to inclusion.
(b) Let g : A > B be a ring homomorphism such that g(s) is a unit in B for all s e S. Show that there exists a unique ring homomorphism h : S_1 A > B such that g = ho f.
2. (a) State and prove the first uniqueness theorem in primary decomposition.
(b) Let K be a field. Show that in the polynomial ring K[xl,x2,x3,...xn] the ideals pi = (x1,x2,x3...xt) (1 < i < n) are prime and all their powers are primary.
3. (a) Let A c B be integral domains, B integral over A. Prove that B is a field iff A is a field.
(b) Let K be a field and B a finitely generated Zf-algebra. If B is a field prove that it is a finite algebraic extension of K.
4. (a) M is a Noetherian A-module <=> every sub module of m is finitely generated - Prove the statement.
(b) Let M be an A-module. If every non empty set of finitely generated submodules of M has a maximal element than prove that M is Noetherian.
5. (a) If A is Noetherian, then prove that the polynomial ring A[x] is Noetherian.
(b) If a finitely generated ring K is a field, prove that it is a finite field.
6. (a) Prove that an Artin ring A is a unique finite direct product of Artin local rings.
(b) Let A be an Artin local ring. Prove the following are equivalent.
(i) every ideal in A is principal.
(ii) the maximal ideal m is principal.
7. (a) Let A be a Noetherian domain of dimension 1. Prove that every non-zero ideal a in A can be uniquely expressed as a product of primary ideals whose radicals are all distinct.
(b) Prove that the ring of integers in an algebraic number field if is a Dedekind domain.
3 MPL-638
|
Attachment: |
| Earning: Approval pending. |
