Abstract Algebra Answers

1 a) Prove that a ring with characteristic zero contains a subring isomorphic to Z.

b) Give an example, with justification, of a ring R which is not a domain, but for which every

deal is principal.

c) Give an example, with justification, of a UFD whose quotient field is C.

2 a) Use the Fundamental Theorem of Homomorphism to prove that Z5/Z2 ~Z3.

1 a) Prove or disprove that we can define a multiplication on R3 = {a+ bi + cj|a,b,c belong to R} such that
i= - 1= j2 .
b) Use the Euclidean algorithm to find a g.c.d. of x5+ x+ 1 and x4+x3+x+1 in Z2[x].

c) If f :R tends to S is a ring homomorphism, then show that char R greater than equal to char S.

Prove that  Δ S4, where   (1 2 3 4).
d) Give an example, with justification, of a ring R which is not a domain, but for which every ideal
is principal.

1) Consider (Z, +) NORMAL (Q, +).

i) Show that every element in Q/Z has finite order, and that there are elements in
Q/Z of order n for any n belong to N.
ii) Show that
Q/Z is an infinite group that is not cyclic.

2a) An isometry of R2 is a map 2 2 :R R which preserves the Euclidean distance between any
two points in R2, i.e., (x)  (y)  x  y x, y R2.
It is known that if (0,0)  (0,0), then  is a linear invertible map, and is called a linear
isometry.
Prove that the set L of linear isometries of R2 is a group with respect to the composition of
functions.
b) Let (G, ) be a group with 5 elements. Construct a Cayley table for . How many distinct
Cayley tables can you construct ?

Which of the following statements are true ? Give reasons for your answers.
i) For any set S, S * S is an equivalence relation.
ii) If H equal to G such that IG:HI = 3, then H is normal in G.
iii) The group of units of ,
13Z
Z with respect to multiplication, is cyclic.
iv) 20 Z is an abelian group with no proper non-trivial normal subgroups (i.e., it is simple).
v) A group of order 168 has either 1 or 8 elements of order 7.
vi) In Z100 there is a unique zero divisor which is also a unit.
vii)
a, b, c R
c 0
a b
is a ring with respect to the usual matrix addition and
multiplication.
viii) Every subring of a commutative ring R is an ideal of R.
ix) Any abelian group is a ring with respect to a suitably defined multiplication.
x) If I and J are ideals in a ring R, then IJ = IJ. (

If A.x=
λx
,where A=

∣ 2 1 -2 ∣
∣ 1 3 1 ∣
∣ 1 2 2 ∣

,determine the eigen values of the matrix A, and an eigen vector corresponding to each eigen value. If λ=2, what is b?

ρ(∂v/∂t + v∙∆v) = -∆p + ∆∙T + f

Where v is the flow velocity, ρ (rho) is the fluid density, p is the pressure, T is the deviatoric component (Cauchy stress tensor), and f represents body force (throughout the volume of the body) and ∆ (which should be upside down) is the del or nabla operator, which denotes the gradient of a vector field.

a) Let V be the vector space of polynomials with real coefficients and of degree at most 2.
If D = d/dx is the differential operator on V and B ={1+2x^2,x+x^2,x^2} is an ordered basis of V,
find [D]B. Find the rank and nullity of D. Is D invertible? Justify your answer.

b) Let T: R^2 →R^2 and S: R^2 →R^2 be linear operators defined by T (x1,x2) = (x1+x2, x1−x2) and S(x1, x2) = (x1, x1+2x2) respectively.
i) Find T◦S and S◦T.
ii) Let B ={(1,0),(0,1)}be the standard basis of R3. Verify that [T◦S]B = [T]B◦[S]B.

c) Find the inverse of the matrix 1 −1 0
2 −1 1
1 1 −1 using row reduction.
d) Let B1 ={(1,1),(1,2)}and B2 ={(1,0),(2,1)}. Find the matrix of the change of basis from B1 to B2.