Past Papers Theory of Modules

[OBJECTIVE]

Subject: Theory of Modules

Time Allowed: 30 Min

Total Marks: 10

NOTE: ATTEMPT THIS PAPER ON THIS QUESTION SHEET ONLY. Division of marks is given in front of each question. This Paper will be collected back after expiry of time limit mentioned above.

 

Part-I Encircle the correct option, each question carries equal marks. (10)

  1. A subset N of an R module M is a sub module of M i f

(a) ax+6y€N

(b) ax+6y€M

(c) ax+6y€R

(d) None of these

  1. Every R- module is isomorphic to a ___________ of a free R-module

(a) Direct summand

(c) isomorphism

(b) quotient module

(d) equivalent

  1. Let A, Band C be sub modules of an R module M such that AC . Then

{a) A⋂(B⋂C) = (A⋂B) ⋂C

(b) A⋂(B+C) = (A⋂B)+C

(c) A+(B⋂C) = (A+B)⋂C

(d) None

  1. According to Dedikind Module law ____________.

(a) (A⋃B)+C = (ANB)+C

(b) A⋃(B+C) = (A⋃(B+C)

(c) A+(B⋃C) = (A+B)⋃C

(d) A+(B⋂C) = (A+B)⋃C

  1. Sub module of C(R) will be

(a) R(I)

(b) I(R)

(c) R(X)

(d) R(R)

  1. If x!=10, y!=0 are elements of a ring R such that xy = 0. Then x and y are called

{a) Multiplicative inverse

(b) Zero Divisor

(c) Additive Inverse

(d) Identity

  1. A mapping fis said to be monomorphism iff fis

(a) Homomorphism

(b) one one

(c) onto

(d) a & b

  1. A root is polynomial equations over the field of rational numbers is called

(a) Integer

(b) Algebraic Number

{c) Rational Integer

(d) Algebraic integer

  1. The identity in quotient R- module M/R = is

{a) M

(b) N

(c} K

(d) 1

  1. If K and L are sub-modules of an R-modute M, then ______________.

(a) (K+L)/K=L/(L⋂K)

(b) (K-L)/K=L/(L⋂X)

(c) (K+L)/K=L/(L⋃K)

(d) (KL)/K=L/(L⋂K)

[SUBJECTIVE]

Subject: Theory of Modules

Time Allowed: 2 Hour 30 Min

Total Marks: 50

NOTE: ATTEMPT THIS (SUBJECTIVE) ON THE SEPARATE ANSWER SHEET PROVIDED

 

Part-II Give short answers, each question carries equal marks. (20)

Q#1: If Mis an irreducible R-modole prove that either M is cyclic or that for every m ∈ M and r ∈ R,rm=0

Q#2: Let M be a module over an integral domain R. Then the set Tof all torsion etements of M is a submodule of M and quotient module M/T is torsion free.

Q#3: Show that every finitely generated R-module is homomorphic image of Free R-module.

Q#4: Every vector space Vover a field F is torsion free F-Module.

Q#5: Show that coefficient of correlation is independent by change of origin and scale.

 

Part-III Give detailed answers, each question carries equal marks. (30)

Q#1: State and prove third isomorphism theorem of modules.

Q#2: Let Rbe aring and M be an R module and f:M->M be a module homomorphism such that f2=f. Show that M =Kerf ® imf

Q#3: Let N be a submodule of an R – module M. show that if N and M/N are FG, then M is FG.