Advanced Abstract Algebra-Ii

Course: M.A.M.Sc. Code: KL-483 Branch: Mathematics Year: 2023

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Total No. of Questions: 11
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KL—483

M. A./M. Sc. Mathematics (Sem. II)

Regular/Private/ATKT/EX

Examination - June - 2023

MATHEMATICS

Paper I

Advanced Abstract Algebra-II

Time : 3 Hours
Maximum Marks : 85

Note : There are three Sections in paper. Attempt all Sections.

Section A 1×10=10

Objective Type Questions

1. Choose the correct answer :

(i) If ring R has unit element 1, then M is called unital R-module, then :

  1. m + 1 = m
  2. 1.m = m
  3. m - 1 = m
  4. (m + 0) = m = (0 + m)

(ii) If A and B are submodules of an R-module M, then (A + B) is :

  1. Submodule of A
  2. Submodule of B
  3. Submodule of M
  4. Linear sum of A and B but not submodule

(iii) Which of the following statements is not correct ?

  1. Any one-dimensional vector space is simple R-module.
  2. A field R regarded as module over itself is a simple R-module.
  3. The only submodules of simple R-module M are {0} and M.
  4. M is simple module if it has proper submodules.

(iv) Which of the following is correct ?

  1. Every ideal Z is free as Z-module.
  2. Every module is homomorphic image of a free module.
  3. Direct product of M1, M2, ......., Mk of free modules is again free.
  4. All of the above are correct

(v) An R-module M is called Artinian of :

  1. descending chain condition hold for M
  2. ascending chain condition hold for M
  3. Both of the above conditions hold for M
  4. None of the above

(vi) Ring of integers is :

  1. Noetherian ring
  2. Artinian ring
  3. Both Noetherian ring and Artinian ring
  4. None of the above

(vii) A non-zero submodule N of a module M is called large, if :

  1. N ∩ K ≠ 0 for all non-zero submodules K of M
  2. N ∩ K = 0 for all non-zero submodules K of M
  3. N ∪ K = K for all non-zero submodules K of M
  4. None of the above

(viii) A module M is called P-primary for same prime ideal P, if :

  1. M has prime ideals.
  2. P is the only prime ideal associated with M.
  3. each non-zero ideal contains same uniform module.
  4. any two prime ideals of are sub-isomorphic.

(ix) The two linear transformations S, T ∈ A(V) are similar, if :

  1. S = T
  2. T = CSC-1 for some C ∈ A(V)
  3. S = |T|-1, where i is identity ∈ A(V)
  4. T = |S|-1, where i is identity ∈ A(V)

(x) The subspace W of V is said to be invariant under T ∈ A(V), if :

  1. WT ⊂ W
  2. WT ⊃ W
  3. WT ⊄ W
  4. None of the above

Section B 5×5=25

Short Answer Type Questions

2. Prove that every Abelian group G is module over the ring of integers.

Or

Prove that intersection of two submodules is also submodule.

3. Define Simple Module and give some examples.

Or

Define Free Module and give some examples.

4. Prove that if M is Noetherian module, then every submodule of M is finitely generated.

Or

Define Artinian module.

5. Define Uniform and Primary modules.

Or

Define prime ideal associated with uniform module.

6. Let A(V) be algebra of linear transformations on V. If T∈A(V), then show that ker T is invariant subspaces of V.

Or

Prove that the relation of similarity is an equivalence relation in A(V).

Section C 5×10=50

Long Answer Type Questions

7. Let M be an R-module, then show the following properties hold :

  1. 0m = 0 ∀m∈M
  2. a0 = 0 ∀a∈R
  3. (-a)m = -(am) = a(-m) ∀a∈R, m∈M

Or

Let f be an R-homomorphism of an R-module M into R-module N, then show that :

M / ker f ≅ img f .

8. Let R be a ring with unity and M be an R-module. Then prove the following statements are equivalent :

  1. M is simple.
  2. M ≠ 0 and M is generated by any 0 ≠ x ∈ M.
  3. M ≅ R/I, where I is maximal left ideal of R.

Or

Let M be a finitely generated unital free R-module with a basis (e₁, e₂, ..., eₙ). Then prove that M ≅ Rⁿ.

9. Let M be an R-module and let N be an R-submodule of M. Then show that M is Noetherian if and only if both N and M/N are Noetherian.

Or

Let R be a noetherian ring. Then show that the polynomial ring is also noetherian ring.

10. Let M be a Noetherian module, then show that each non-zero submodule contains a uniform module.

Or

Let M be a non-zero finitely generated module over a commutative Noetherian ring R. Then show that there are only a finite number of primes associated with M.

11. If V is finite-dimensional over F, then show that T∈A(V) is invertible if and only if the constant term of the minimal polynomial for T is not zero.

Or

Let V be a n-dimensional vector space over field F and T∈A(V) has all its characteristic root in F. Then show that there is a basis of V in which matrix of T is triangular.

KL-483
P.T.O.
(8)