MTH721 - Commutative Algebra

[image: kNty4oI.png] MTH721 (Spring 2020) Assignment No. 1 Maximum Marks: 25 Due Date: May 31, 2020 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 : ab=〖(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.

MTH721 (Spring 2020) Assignment No. 2 Maximum Marks: 15 Due Date: June 14, 2020 INSTRUCTIONS Please read the following instructions before attempting the solution of this assignment: • To solve this assignment, you should have good command over 8 to 12 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 [image: LyHiOm6.png] Question: 2 Marks: 5 [image: xvaU3UB.png] Question: 3 Marks: 5 [image: zXa0RnF.png] Spring 2020_MTH721_2.pdf

MTH721 (Fall 2019) Assignment No. 4 Maximum Marks: 20 Due Date: February 02, 2020 INSTRUCTIONS Please read the following instructions before attempting the solution of this assignment: • To solve this assignment, you should have good command over 29 to 35 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: 10 Compute the reduced Grobner basis of the ideal [image: PPUG7fO.png] using the lexicographic order . [image: uF4drHi.png] Question: 2 Marks: 10 Compute the reduced Grobner basis of the ideal [image: vtUJdOs.png] using the lexicographic order . [image: ijSeXRl.png]

MTH721 (Fall 2019) Assignment No. 3 Maximum Marks: 10 Due Date: January 12, 2020 INSTRUCTIONS Please read the following instructions before attempting the solution of this assignment: • To solve this assignment, you should have good command over 17 to 19 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: 10 Write the ideal [image: obqqm7R.png]as a finite intersection of primary ideals.