Abstract Algebra Answers

a) Give two distinct maximal ideals of R[x] , with justification.
b) Give an example, with justification, of rings R and S with a ring homomorphism
φ:R→S such that P is a prime ideal of R but φ(P) is not a prime ideal of S.
c) Let D be a Euclidean domain, with Euclidean valuation d . Prove that if n∈N such
that d(1) + n≥0 , then f :D \ {0}→Z : f (x) = d(x) + n is also a Euclidean valuation
on D .

a) Consider the ring R= Z(subscript)3[x] / < x^8 - 1>
i) Is R a finite ring?
ii) Does R have zero divisors?
iii) Does R have nilpotent elements?
Give reasons for your answers.
b) Give an example, with justification, of a subset of a ring that is a subgroup under
addition but not a subring.
c) Construct a multiplication (Cayley) table for 3Z/9Z.

a) If G is a group of order 40, and H and K are its subgroups of orders 20 and 10,
then check whether or not HK ≤ G . Further, show that o(H∩K) ≥ 5.
b) Prove that C*/S~=R+, where S ={z∈C*| |z|=1}, R+ ={x∈R| x>0} and C*=C\{0}.
c) What are the possible algebraic structures of a group of order 99?

a)Let S be a set with n elements, n >= 3 . Let B be the set of bijective mappings of S
onto itself.
i) Check whether (B, o) is a group or not.
ii) Give the cardinality of the set B .
iii) Is o commutative? Give reasons for your answer.
b) Check whether or not H = {x∈R*|x =1 or x∉Q } is a subgroup of ( R*,.) . Also
check whether K = {x∈R*|x>=1} is a subgroup of R* or not.
c) Let U(n) ={x∈N|1<+x<n,(x,n)=1}. Show that U(n) is a group w.r.t.
multiplication modulo n . Also show that U(14) is cyclic and U(20) is not
cyclic.
d) Obtain the centre of Q_8 and two distinct right cosets of Z(Q_8 ) in Q_8.

1. Which of the following statements are true? Give reasons for your answers.
i) If f : S->S is 1-1, where S is a set, then f is onto.
ii) The signature of the product of k disjoint cycles in S_n is k .
iii) R[x] has no maximal ideal.
iv) If m| n in Z , then mZ is an ideal of Z/ nZ .
v) Every non-trivial subgroup of an infinite group is infinite.
vi) The characteristic of the quotient field of any UFD is zero.
vii) If H and K are subgroups of a group G , then HK is also a subgroup of G .
viii) If the order of a group is even, then it cannot have an element of odd order.
ix) If R is an integral domain, then R / I has no zero divisors, where I is an ideal of R .

Let d€N, where d not equal to 1 and d is not divisible by the square of a prime. Prove that N:Z[√d] such that N union {0} : N(a+b√d)=|a^2-db^2| satisfies the following properties for x,y € Z[√d]:
1) N(x)=0 that implies x=0
2) N(x)=1 implies that x is a unit
3) N(x) is prime implies that x is irreducible in Z[√d]

Let d∈N , where d ≠1 and d is not divisible by the square of a prime. Prove that
N:Z[ d root(2)]→N∪{0}:N(a+b root(d))=|a^2 −db^2 | satisfies the following properties for x,y∈Z[ droot(2)]:
i) N(x) = 0 ⇔ x = 0
ii) N(xy) = N(x)N(y)
iii) N(x)=1⇔x is a unit
iv) N(x) is prime ⇒ x is irreducible in Z[ d ] .

If ω be the imaginary cube root of unity show that the set {1,w,w^2)is a cyclic group of order 3 with respect to multiplication.

Show that (2,X) is maximal in Z+XQ[X].

Verify that K[[X]] is a local ring,where K is a field