Abstract Algebra Answers

Find the permutation in S_4
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such that 1→3, 2 12→1, 23→2.

Calculate 2+2 in Z_3

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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 ?

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.

In following statements identify if they are true then give proof or if they are false then give counter example or if they are partially true then give example for each case in which it is true and false.
1. Minimal Modules exist in Artinian Modules ?
2. A Minimal Sub-Module is always Simple ?
3. A Finitely Generated Module need not be Noetherian ?
4. An Artinian Module need not be Finitely Generated ?

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.