Abstract Algebra Answers

modules which are neither Noetherian nor Artinian.is it true of false.if it is one them then give proof

Why zero ring is artinian as well as noetherian?

A Minimal Sub-Module is always Simple ?

Minimal Modules exist in Artinian Modules ?

Give a non trivial element of the ring Z10 / <4bar> , with justification.

(1) any two groups of order m are isomorphic where m€ N

(2)if A and B are two sets such that A U B = phi , then A intersection B = phi

True/false. Justify.

(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