Advanced Abstract Algebra-Ii

(i) For an R-module M, trivial submodules are

(a) O

(b) M

(c) both a and b

(d) None of these

(ii) An R-module M is called cyclic if for some X∈M

(a) M = (x)

(b) M ≠ (x)

(c) M = (x²)

(d) None of these

(iii) An R-module M is called simple if

(a) R/M = (0)

(b) R/M ≠ 0

(c) RM = (0)

(d) RM ≠ (0)

(iv) Let M be a finitely generated free module over a commutative ring R. Then all bases of M are

(a) finite

(b) infinite

(c) finite or infinite

(d) None of these

(v) A module which has only finitely many submodules is

(a) noetherian

(b) artinian

(c) noetherian and artinian both

(d) None of these

(vi) Let R be a noetherian ring. Then the polynomial ring R[x] is also a noetherian ring is known as

(a) Schur's lemma

(b) Wedderbun-Artin theorem

(c) Noether-Loskar theorem

(d) Hilbert basis theorem

(vii) A non zero module M is called uniform if any two nonzero submodules of M have

(a) zero intersection

(b) non zero intersection

(c) zero union

(d) non zero union

(viii) Z as a Z-module is

(a) uniform module

(b) primary module

(c) uniform as well as primary module

(d) None of these

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

(a) WT ⊂ W

(b) WT ⊃ W

(c) WT = W

(d) None of these

(x) If T∈A (V) then λ∈F is called an eigen value of T if λI − T is

(a) singular

(b) nonsingular

(c) singular and nonsingular both

(d) None of these

2. Let A be any additive abelian group. Then prove A is a left Z-module.

OR

If M is an R-module and X ∈ M, then prove that the set Rx = {rx | r∈R} is an R-submodule of M.

3. Define simple module and semisimple module.

OR

Prove that the set {1, x, 1+x, x², x³} in F-module F (x) is linearly dependent.

4. Define Artinian Module with example.

OR

Prove that every homomorphic image of a noetherian module is noetherian.

5. Write the statement of Noether Lasker Theorem.

OR

Define primary module and P-primary module.

6. Prove that the element λ∈F is a characteristic root of T∈A (V) iff for some v ≠ 0 in V, VT = λV.

OR

Define similarity of linear transformations.

7. Define submodule and let {N_i}∈A be a family of R-submodules of an R-module M. Then prove that ∩N_i is also an R-submodule.

OR

Let f be an R-homomorphism of an R-module M into an R-module N. Then prove M/Kerf ≅ f(M).

8. Let V be a nonzero finitely generated vector space over a field F. Then prove that V admits a finite basis.

OR

State and prove Schur's lemma.

9. For an R-module M, if every submodule of M is finitely generated, then prove that every nonempty set S of submodules of M has a maximal elements.

OR

Let R be a noetherian ring. Then prove that the polynomial ring R[x] is also a noetherian ring.

10. Let M be a noetherian module or any module over a noetherian ring. Then prove that each nonzero submodule contains a uniform module.

OR

State and prove fundamental structure theorem of modules over a principal ideal domain.

11. Prove that the minimal polynomial of a linear operator T∈A (V) divides its characteristic polynomial.

OR

Let V be n-dimensional over F and let T∈A (V) has all its characteristic roots in F. Then prove that T satisfies a polynomial of degree n over F.