MTH721 (Spring 2020) Assignment No. 1

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Maximum Marks: 25
Due Date: May 31, 2020
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INSTRUCTIONS

Please read the following instructions before attempting the solution of this assignment:

• To solve this assignment, you should have good command over 1 to 6 Lectures.

• Try to get the concepts, consolidate your concepts which you learn in these lectures with

these questions.

• Upload assignments properly through LMS. No Assignment will be accepted through

email.

• Write your ID on the top of your solution file.

Do not use colorful backgrounds in your solution files.

Use Math Type or Equation Editor etc. for mathematical symbols and equations.

Zero marks will be awarded for a copied solution. That is if the solution files of any two students are found same, both of them will be awarded zero marks. Therefore, try to make solution by yourself and protect your work from other students.

Avoid copying the solution from book (or internet); you must solve the assignment yourself.

Also remember that you are supposed to submit your assignment in Word format any other format like scanned images, HTML etc. will not be accepted

Note: Attempt all the following questions.

Question: 1 Marks: 5

Determine whether the binary operation * defined by *:R×R→R and given for all a,b∈R as : a*b=〖(a+b)〗^2 is associative or not? Explain your answer.

Question: 2 Marks: 5

Show that C, the set of all non-zero complex numbers is a multiplicative group.

Question: 3 Marks: 5

Show that the following function f:Z_2→Z_2 is a ring homomorphism:

f(x)=x^2

Question: 4 Marks: 5

Show that the following function g:Z→Z is not a ring homomorphism:

f(x)=2x

Question: 5 Marks: 5

Show that in a principal ideal domain, every nonzero prime ideal is maximal.