Abstract Algebra Answers

Check whether or not S={a0+a1x+.......+anx^n belongs to Z[x]|5|a0}is an ideal of Z[x]

Let G be a group and H be a non-empty finite subset of .G If ,H ab∈H∀ b,a ∈
then show that H ≤ .G Will the result remain true if H is not finite? Give reasons
for your answer.

Show that if G is a non-cyclic group of order n, then G has no element of order
n. Further, give an example, with justification, of a non-cyclic group with all its
proper subgroups being cyclic.

Let R be a ring. Show that M3(R) is a ring with respect to the usual matrix
addition and multiplication. Further, if R is commutative, will M3(R )be
commutative? Why, or why not?

Consider the ideal I<x³-1.x⁴+2x³+7x²+5x+5> in Q[x]. Find p(x) ∈Q[x] such that I =<p(x)> .
Is Q[x]/I a field? Give reasons for your answer.

Which of the following statements are true? Give reasons for your answers, in the form
of a short proof or a counterexample.
i) M3 (Z) has no nilpotent elements.
ii) If P and 1 P are prime ideals of a ring 2
,R then . P1 P2 = P1 ∩ P2
iii) The set of cosets of <(1 2)> in 3S is a group with respect to multiplication of cosets.
iv) If (G,*) is a group, then , * is the only binary operation defined on .G
v) If every element of a group G has finite order, then G must be of finite order.
vi) If k is a field, so is k × k.
vii) x is a unit in R[x].
viii) If A and B are two sets such that A∪B = fi then A ∩B =fi.
ix) Q[x]/<x^6+17> is a field of characteristic 6.
x) Any two groups of order m are isomorphic, where m∈N.

Prove that R^5/R=R⁴ as rings.

Show that if :f Q->Q is a ring homomorphism, then f(x) = x ∀x ∈ Q. Would
this still be true if f were a group homomorphism? Why, or why not?

Show that Z[√-2] is not a UFD.

Z ring of integers, R ring, f: ring homomorphism from Z to R which is injective. Also if n is not in 13Z, then f(n) is unit in R. Can R be integral domain. One example of such a map.