Abstract Algebra Answers

In each of the following statement, identify if they are true, false or partially true. In case it is true, give proof. In case it is false give counter example with solution. In case it is partially true then give one example each of the case in which it is true and in which it is not true.

1. If R is Artinian, then R[x] is Artinian.
2. A finitely generated module over a PID is Noetherian.
3. Minimal modules exist in Noetherian modules.
4. Maximal Submodule need not exist in an Artinian module.
5. A Noetherian module need not be Artinian.
6. A maximal submodule is always simple.

(1) if P1 and P2 are prime ideals of a ring R , then P1P2 = P1 intersection P2.
(2) if k is a field , so is k × k
(3) Q[x]/<x^6 + 17 > is a field of characteristic 6.
True/false. Justify.

Let R be a ring for which ab = ca implies b = c for all a,b,c belongs to R, a not equal to zero. Show that R is commutative.

Show that d : Q[x]\{0} -> N U {0} : d(f) = 5^deg f is a Euclidean valuation on Q[x]

Use the ring Z(underroot -2) to show that (1) the quotient ring of a ufd need not be a UFD. (2)an irreducible element of a UFD need not be a prime element

Use the ring Z(underroot -2) to show that (1) the quotient ring of a ufd need not be a UFD. (2)an irreducible element of a UFD need not be a prime element

Which of these is not a set of numbers
a.complex
b.intregral domain
c.real
d.natural

x is a unit in R[x]

1.If (G,*)is a group, then , * is the only binary operation defined on G

2.If every element of a group G has finite order, then G must be of finite order.

The set of cosets of <(1 2)> in S3 is a group with respect to multiplication of cosets.