Advance Abstract Algebra-Ii

Which of the following are not R-module ?
(a) Any Abelian group G over ring of integer R
(b) Ring R itself over ring R
(c) Polynomial Ring R[x] over ring R
(d) None of the above

An R-module M is called cyclic module if :
(a) M = {x, y, z] for some x, y, z ∈ M
(b) M = {x₁, x₂,...,x_n} for x₁, x₂,...,x_n ∈ M.
(c) M = ∑^m_n=1 a_nx_n where a_n ∈ R and x_n ∈ M!
∀i = 1,2,...n
(d) M = {x} for some x ∈ M

A homomorphism f : M → N of R-module M and N is called an automorphism :
(a) If f is one-one onto
(b) If f is one-one into
(c) If M = N
(d) If M = N and f is one-one onto

Which is not true about free module over ring R ?
(a) Rⁿ is a free R-module if R has unity
(b) Direct sum of free module is a free module
(c) An R-module M is called free module if M does not have basis set
(d) None of the above

An R-module M is called Noetherian if :
(a) Descending chain condition holds for M
(b) Ascending chain condition holds for M
(c) Ascending and Descending conditions are not true for M
(d) M is infinite dimensional vector space

Which of the following is a correct statement ?
(a) A noetherian module is always artinian
(b) A finitely generated module is always noetherian
(c) An artinian module is always finitely generated
(d) Minimal submodules need not exist in a noetherian module

A non zero submodule N of a module M is called large if :
(a) N∩K = N ∀ non zero submodules K of M
(b) N∩K = K ∀ non zero submodules K of M
(c) N∩K ≠ 0 ∀ non zero submodules K of M
(d) N∩K = 0 ∀ non zero submodules K of M

A module M is called P-primary for some prime ideal P if :
(a) P is the only prime ideal associated with M
(b) P is not the only prime ideal associated with M
(c) Each non-zero submodule of P has uniform module
(d) Any two uniform submodules of P are sub-isomorphic

The two linear transformations S, T ∈ A(V) are said to be similar if :
(a) T = CSC^-1 for some C ∈ A(V)
(b) T = CS for some C ∈ A(V)
(c) T = SCS^-1 for some C ∈ A(V)
(d) T = S for some C ∈ A(V)

Let U be a vector space over F of dimension m. Then Hom(U, U) is a vector space over F of dimension :
(a) m
(b) m + m
(c) m · m
(d) Infinite

Let M be an R-module. Then prove the following properties :
(i) om = 0 ∀m ∈ M where o ∈ R and O ∈ M

(ii) aO = 0 ∀a ∈ R where O ∈ M.

Or
Prove that intersection of two submodules is also a submodule.

Define simple R-modules and give any two examples.

Or
Define Free module and Rank of Free module.

Prove that an R-module M is noetherian if every submodule of M is finitely generated.

Or
Define artinian module and give any example.

Define Prime Ideal associated with uniform module.

Or
Prove that subisomorphism is an equivalence relation.

If V is finite-dimensional over F then show that T ∈ A(V) is invertible if and only if constant term of the minimal polynomial for T is not 0.

Or
Prove that the element λ ∈ F is characteristic root of T ∈ A(V) if and only if for some v ≠ 0 in V vT = λv.

Let (N_i)_i∈A be a family of R-submodules of R-module M. Then show that the following are equivalent :
(i) ∑_i∈A N_i is a direct sum

(ii) 0 = ∑_i∈A x_i ∈ ∑_i∈A N_i implies x_i = 0 for all i ∈ A

(iii) ∑_{j∈A, j≠i} N_j = (0) ∀i ∈ A

Or
Prove that the submodules of the quotient module M/N are of the form U/N, where U is submodule of M containing N.

Let R be a ring with unity and let M be an R-module. Then show that the following statements are equivalent :
(i) M is simple

(ii) M ≠ 0 and M is generated by any 0 ≠ x ∈ M

(iii) M = R/I where I is a maximal left ideal of R

Or
Let M be a free R-module with basis set {e₁, e₂,...,e_n}. Then show M ≅ Rⁿ.

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

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

Let M be a noetherian module. Then show that each non-zero submodule contains a uniform module.

Or
Let U be a uniform module over a commutative noetherian ring R. Then show that U contains a submodule isomorphic to R/P for precisely one prime ideal P, that is U is subisomorphic to R/P for some exactly one prime ideal P.

If W ⊆ V be invariant under T ∈ A(V) then show that T induces a linear transformation &overline{T} on the quotient space V/W defined by (v + W) &overline{T} = vT + W. Further if T satisfies the polynomial q(x) ∈ F[x] then show &overline{T} also satisfies q(x) and if P₁(x) is the minimum polynomial for &overline{T} over F and if P(x) is that for T then show P₁(x)|P(x).

Or
Let V be an n-dimensional vector space over a field F and let T ∈ A(V) has all its characteristic roots on F then show that there is basic of V in which the matrix of T is triangular.