Answer to Question #107002 in Abstract Algebra for Kamalpreet Kaur

Question #107002

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.

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Assignment Expert

02.04.20, 00:43

Additional assumptions were not given, but there is a statement which relates two terms. For example, if M is a maximal left ideal of a ring R, then R/M is a simple R-module.

Irshad

01.04.20, 08:38

A simple module is always maximal.Is it true or false.If it is one of them then give froof

Assignment Expert

30.03.20, 13:41

Dear Sanjana. Please use the panel for submitting new questions.

Sanjana

30.03.20, 10:17

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 ?

Answer to Question #107002 in Abstract Algebra for Kamalpreet Kaur

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