
MTH721 (Spring 2020) Assignment No. 1
Maximum Marks: 25 Due Date: May 31, 2020INSTRUCTIONS
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 nonzero 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
Question: 2 Marks: 5
Question: 3 Marks: 5
Spring 2020_MTH721_2.pdf 
MTH721 (Fall 2019) Assignment No. 4
Maximum Marks: 20
Due Date: February 02, 2020INSTRUCTIONS
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
using the lexicographic order .
Question: 2 Marks: 10
Compute the reduced Grobner basis of the ideal
using the lexicographic order . 
MTH721 (Fall 2019) Assignment No. 3
Maximum Marks: 10
Due Date: January 12, 2020INSTRUCTIONS
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 as a finite intersection of primary ideals.
MTH721 Assignment 1 Solution and Discussion

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 nonzero 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. 
@cyberian said in MTH721 Assignment 1 Solution and Discussion:
Question: 5 Marks: 5
Show that in a principal ideal domain, every nonzero prime ideal is maximal. 
@cyberian said in MTH721 Assignment 1 Solution and Discussion:
Question: 4 Marks: 5
Show that the following function g:Z→Z is not a ring homomorphism:
f(x)=2x 
@cyberian said in MTH721 Assignment 1 Solution and Discussion:
Question: 3 Marks: 5
Show that the following function f:Z_2→Z_2 is a ring homomorphism:
f(x)=x^2 
@cyberian said in MTH721 Assignment 1 Solution and Discussion:
Question: 2 Marks: 5
Show that C, the set of all nonzero complex numbers is a multiplicative group.Answer:
Let C={z:z=x+iy, x,y∈R}C={z:z=x+iy, x,y∈R}. Here R is the set of all real numbers and i=√(1).
(G1) Closure Axiom: If a+ib∈C and c+id∈C, then by the definition of multiplication of complex numbers
(a+ib)(c+id)=(ac–bd)+i(ad+bc)∈C
Since ac–bd,ad+bc∈R, for a,b,c,d∈R. Therefore,C is closed under multiplication.
(G2) Associative Axiom:
(a+ib){(c+id)(e+if)}=(ace–adf–bcf–bde)+i(acf+ade+bce–bdf)
={(a+ib)(c+id)}(e+if) for a,b,c,d∈R .
(G3) Identity Axiom: e=1(=1+i0) is the identity in C.
(G4) Inverse Axiom: Let (a+ib)(≠0)∈C, then
(a+ib)^(1)=1/(a+ib)=(aib)/(a^2+b^2 )
=a/(a^2+b^2 )i b/(a^2+b^2 )=m+in∈∁
Hence C is a multiplicative group. 
@cyberian said in MTH721 Assignment 1 Solution and Discussion:
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.Answer:
Consider the elements 1,3,6∈R. Then we have that:1∗(2∗3)=1∗(2+3)2=1∗25=(1+25)2=676
We also have that:(1∗2)∗3=(1+2)2∗3=9∗3=(9+3)2=122=144
Clearly 676≠144 and so ∗ is nonassociative on R since a∗(b∗c)≠(a∗b)∗c for 1,3,6∈R.