Rings | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
2. Rings 
2.1 How many proper, non-zero ideals does the ring Z
????
 have? Justify your 
answer. How many ideals does the ring Z
????
?Z
????
 have? Why? 
(???????? :?? +?? +?? +?? =???? Marks) 
Solution: 
Let (Z
?? ,?
?? ) be a cyclic group of order ?? . 
Let ?? be a subgroup of Z
?? . 
? 
?? =(?? )=???? , f.s. ?? ?Z 
We will show ?? is an ideal of Z
?? . 
Let ?? ,?? ??? . 
????????????????????????????????????????????????????????????????? ?=?? 1
?? and ?? =?? 2
?? ????????????????????????????????????????(??) 
So,  
????????????????????????????????????????????????????????????? -?? =?? 1
?? -?? 2
=(?? 1
-?? 2
)?? ??? ?????????????????????????(???? )???? 
and let ?? ??? ;?? ?Z
?? . 
Consider 
?? =?? 1
?? (?? 1
)?? ??? (???? ) 
From (i) and (ii) 
?? is an ideal of Z
?? . 
And number of subgroup of Z
?? =?? (?? ) 
??            Number of ideal of Z
?? =?? (?? ) 
??            Number of ideal of Z
12
=?? (12) 
=?? (3.4)=4 
? Number of proper non-zero ideal of Z
12
 
=4-2=2 
Page 2


Edurev123 
2. Rings 
2.1 How many proper, non-zero ideals does the ring Z
????
 have? Justify your 
answer. How many ideals does the ring Z
????
?Z
????
 have? Why? 
(???????? :?? +?? +?? +?? =???? Marks) 
Solution: 
Let (Z
?? ,?
?? ) be a cyclic group of order ?? . 
Let ?? be a subgroup of Z
?? . 
? 
?? =(?? )=???? , f.s. ?? ?Z 
We will show ?? is an ideal of Z
?? . 
Let ?? ,?? ??? . 
????????????????????????????????????????????????????????????????? ?=?? 1
?? and ?? =?? 2
?? ????????????????????????????????????????(??) 
So,  
????????????????????????????????????????????????????????????? -?? =?? 1
?? -?? 2
=(?? 1
-?? 2
)?? ??? ?????????????????????????(???? )???? 
and let ?? ??? ;?? ?Z
?? . 
Consider 
?? =?? 1
?? (?? 1
)?? ??? (???? ) 
From (i) and (ii) 
?? is an ideal of Z
?? . 
And number of subgroup of Z
?? =?? (?? ) 
??            Number of ideal of Z
?? =?? (?? ) 
??            Number of ideal of Z
12
=?? (12) 
=?? (3.4)=4 
? Number of proper non-zero ideal of Z
12
 
=4-2=2 
Using the results, "If ?? is an ideal of ring ?? and ?? is an ideal of ring ?? then ?? ×?? is a ideal 
of ?? ×?? ." 
? Number of ideals of 
Z
12
×Z
12
?=?? (12)×?? (12)
?=4×4=16
 
2.2 Show that Z[?? ] is a unique factorization domain that is not a principal ideal 
domain ( Z is the ring of integers). Is it possible to give an example of principal 
ideal domain that is not a unique factorization domain? Z[?? ] is the ring of 
polynomials in the variable ?? with integers). 
(2009 : 15 Marks) 
Solution: 
We have to show Z[?? ] is U.F.D. 
As we know, if ?? is U.F.D., then ?? (?? ) is U.F.D 
We know, every P.I.D. is U.F.D. 
?Z is P.I.D. 
?Z is U.F.D. 
?Z[?? ] is U.F.D. 
in PI.D.; ?? ?0 and non-unit is irreducible 
?<?? > is maximal. 
In Z[?? ],?? is irreducible. 
?? =?? .?? 
Here we have two possibilities, ?? is constant and ?? is of degree 1 , or ?? is constant and ?? 
is constant and ?? is of degree 1 . 
Case 1:?? is constant, ?? is of degree 1 . 
?? ?=?? (???? +?? )
?=?????? +????
 
On comparing co-efficient, 
???? =1 and ???? =0
???? =1 and ?? =0?(??? is constant )
 
Page 3


Edurev123 
2. Rings 
2.1 How many proper, non-zero ideals does the ring Z
????
 have? Justify your 
answer. How many ideals does the ring Z
????
?Z
????
 have? Why? 
(???????? :?? +?? +?? +?? =???? Marks) 
Solution: 
Let (Z
?? ,?
?? ) be a cyclic group of order ?? . 
Let ?? be a subgroup of Z
?? . 
? 
?? =(?? )=???? , f.s. ?? ?Z 
We will show ?? is an ideal of Z
?? . 
Let ?? ,?? ??? . 
????????????????????????????????????????????????????????????????? ?=?? 1
?? and ?? =?? 2
?? ????????????????????????????????????????(??) 
So,  
????????????????????????????????????????????????????????????? -?? =?? 1
?? -?? 2
=(?? 1
-?? 2
)?? ??? ?????????????????????????(???? )???? 
and let ?? ??? ;?? ?Z
?? . 
Consider 
?? =?? 1
?? (?? 1
)?? ??? (???? ) 
From (i) and (ii) 
?? is an ideal of Z
?? . 
And number of subgroup of Z
?? =?? (?? ) 
??            Number of ideal of Z
?? =?? (?? ) 
??            Number of ideal of Z
12
=?? (12) 
=?? (3.4)=4 
? Number of proper non-zero ideal of Z
12
 
=4-2=2 
Using the results, "If ?? is an ideal of ring ?? and ?? is an ideal of ring ?? then ?? ×?? is a ideal 
of ?? ×?? ." 
? Number of ideals of 
Z
12
×Z
12
?=?? (12)×?? (12)
?=4×4=16
 
2.2 Show that Z[?? ] is a unique factorization domain that is not a principal ideal 
domain ( Z is the ring of integers). Is it possible to give an example of principal 
ideal domain that is not a unique factorization domain? Z[?? ] is the ring of 
polynomials in the variable ?? with integers). 
(2009 : 15 Marks) 
Solution: 
We have to show Z[?? ] is U.F.D. 
As we know, if ?? is U.F.D., then ?? (?? ) is U.F.D 
We know, every P.I.D. is U.F.D. 
?Z is P.I.D. 
?Z is U.F.D. 
?Z[?? ] is U.F.D. 
in PI.D.; ?? ?0 and non-unit is irreducible 
?<?? > is maximal. 
In Z[?? ],?? is irreducible. 
?? =?? .?? 
Here we have two possibilities, ?? is constant and ?? is of degree 1 , or ?? is constant and ?? 
is constant and ?? is of degree 1 . 
Case 1:?? is constant, ?? is of degree 1 . 
?? ?=?? (???? +?? )
?=?????? +????
 
On comparing co-efficient, 
???? =1 and ???? =0
???? =1 and ?? =0?(??? is constant )
 
??? is unit. 
Similarly in Case II ?? is unit. 
??? is irreducible. 
As ??? ????? ,2??[?? ]. But ??? ? is not maximal 
From ?
*
),??? ? is not maximal but ?? is not irreducible. 
?Z[?? ] is not P.I.D. 
Further, no it is not possible to give an example of P.I.D. that is not U.F.D. because 
every P.I.D. is U.F.D. 
2.3 How many elements does the quotient ing 
Z
?? [?? ]
(?? ?? +?? )
 have? Is it an integral 
domain? Justify your answer. 
(2009 : 15 Marks) 
Solution: 
 Let 
?? ?=
Z
5
[?? ]
??? 2
+1?
?? ?=
Z
5
[?? ]
??? 2
+1?
={?? (?? )+??? 2
+1?:?? (?? )?Z
5
(?? )}
 
By division algorithm on ?? (?? ) and ?? 2
+1 
??? (?? ) and ?? (?? ) 
?? (?? )?=?? (?? )(?? 2
+1)+?? (?? ); where ?? (?? )=0
deg??? (?? )?<deg?(?? 2
+1)
?? (?? )+??? 2
+1??=?? (?? )??? 2
+1?+?? (?? )+??? 2
+1?
?=?? (?? )??? 2
+1?+??? 2
+1?+?? (?? )+??? 2
+1?
?=??? 2
+1?+?? (?? )+??? 2
+1?
?=?? (?? )+??? 2
+1?
???? ?={?? (?? )+??? 2
+1?:?? (?? )?Z
5
[?? ]deg??? (?? )=1}
?={?? +???? +??? 2
+1?:?? ,?? ?Z
5
 
Thus, we have five choices for ' ?? ' as well as five choices for ' ?? '. 
??????????????????????????????????????????????????|?? |=5×5=25 
As ?? 2
+1 is reducible over Z
5
, because 
?? 2
+1=(?? +2)(?? +3) 
? By Chinese Remainder theorem; 
Page 4


Edurev123 
2. Rings 
2.1 How many proper, non-zero ideals does the ring Z
????
 have? Justify your 
answer. How many ideals does the ring Z
????
?Z
????
 have? Why? 
(???????? :?? +?? +?? +?? =???? Marks) 
Solution: 
Let (Z
?? ,?
?? ) be a cyclic group of order ?? . 
Let ?? be a subgroup of Z
?? . 
? 
?? =(?? )=???? , f.s. ?? ?Z 
We will show ?? is an ideal of Z
?? . 
Let ?? ,?? ??? . 
????????????????????????????????????????????????????????????????? ?=?? 1
?? and ?? =?? 2
?? ????????????????????????????????????????(??) 
So,  
????????????????????????????????????????????????????????????? -?? =?? 1
?? -?? 2
=(?? 1
-?? 2
)?? ??? ?????????????????????????(???? )???? 
and let ?? ??? ;?? ?Z
?? . 
Consider 
?? =?? 1
?? (?? 1
)?? ??? (???? ) 
From (i) and (ii) 
?? is an ideal of Z
?? . 
And number of subgroup of Z
?? =?? (?? ) 
??            Number of ideal of Z
?? =?? (?? ) 
??            Number of ideal of Z
12
=?? (12) 
=?? (3.4)=4 
? Number of proper non-zero ideal of Z
12
 
=4-2=2 
Using the results, "If ?? is an ideal of ring ?? and ?? is an ideal of ring ?? then ?? ×?? is a ideal 
of ?? ×?? ." 
? Number of ideals of 
Z
12
×Z
12
?=?? (12)×?? (12)
?=4×4=16
 
2.2 Show that Z[?? ] is a unique factorization domain that is not a principal ideal 
domain ( Z is the ring of integers). Is it possible to give an example of principal 
ideal domain that is not a unique factorization domain? Z[?? ] is the ring of 
polynomials in the variable ?? with integers). 
(2009 : 15 Marks) 
Solution: 
We have to show Z[?? ] is U.F.D. 
As we know, if ?? is U.F.D., then ?? (?? ) is U.F.D 
We know, every P.I.D. is U.F.D. 
?Z is P.I.D. 
?Z is U.F.D. 
?Z[?? ] is U.F.D. 
in PI.D.; ?? ?0 and non-unit is irreducible 
?<?? > is maximal. 
In Z[?? ],?? is irreducible. 
?? =?? .?? 
Here we have two possibilities, ?? is constant and ?? is of degree 1 , or ?? is constant and ?? 
is constant and ?? is of degree 1 . 
Case 1:?? is constant, ?? is of degree 1 . 
?? ?=?? (???? +?? )
?=?????? +????
 
On comparing co-efficient, 
???? =1 and ???? =0
???? =1 and ?? =0?(??? is constant )
 
??? is unit. 
Similarly in Case II ?? is unit. 
??? is irreducible. 
As ??? ????? ,2??[?? ]. But ??? ? is not maximal 
From ?
*
),??? ? is not maximal but ?? is not irreducible. 
?Z[?? ] is not P.I.D. 
Further, no it is not possible to give an example of P.I.D. that is not U.F.D. because 
every P.I.D. is U.F.D. 
2.3 How many elements does the quotient ing 
Z
?? [?? ]
(?? ?? +?? )
 have? Is it an integral 
domain? Justify your answer. 
(2009 : 15 Marks) 
Solution: 
 Let 
?? ?=
Z
5
[?? ]
??? 2
+1?
?? ?=
Z
5
[?? ]
??? 2
+1?
={?? (?? )+??? 2
+1?:?? (?? )?Z
5
(?? )}
 
By division algorithm on ?? (?? ) and ?? 2
+1 
??? (?? ) and ?? (?? ) 
?? (?? )?=?? (?? )(?? 2
+1)+?? (?? ); where ?? (?? )=0
deg??? (?? )?<deg?(?? 2
+1)
?? (?? )+??? 2
+1??=?? (?? )??? 2
+1?+?? (?? )+??? 2
+1?
?=?? (?? )??? 2
+1?+??? 2
+1?+?? (?? )+??? 2
+1?
?=??? 2
+1?+?? (?? )+??? 2
+1?
?=?? (?? )+??? 2
+1?
???? ?={?? (?? )+??? 2
+1?:?? (?? )?Z
5
[?? ]deg??? (?? )=1}
?={?? +???? +??? 2
+1?:?? ,?? ?Z
5
 
Thus, we have five choices for ' ?? ' as well as five choices for ' ?? '. 
??????????????????????????????????????????????????|?? |=5×5=25 
As ?? 2
+1 is reducible over Z
5
, because 
?? 2
+1=(?? +2)(?? +3) 
? By Chinese Remainder theorem; 
Z
5
[?? ]
??? 2
+1?
=
Z
5
[?? ]
??? +2???? +3?
?
Z
5
[?? ]
??? +2?
×
Z
5
[?? ]
??? +3?
(?g.c.d?(?? +2),(?? +3)=1) 
?Z
5
 is field and ?? +2 is irreducible polynomial over Z
5
. 
?
Z
5
[?? ]
??? +2?
 is field. 
And 
Z
5
[?? ]
??? +2?
={?? 0
+??? +2?:?? 0
?Z
5
} 
?????????????????????????????????????????|
Z
5
[?? ]
??? +2?
|=5 
????????????????????????????????????????????
Z
5
[?? ]
??? +2?
?Z
5
 
Similarly, 
Z
5
[?? ]
??? +3?
?Z
5
 
?????????????????????????????????????????????????????????????????????????
Z
5
[?? ]
??? +1?
?Z
5
×Z
5
 
?? =(1,0)?Z
5
×Z
5
 
?? =(0,1)?Z
5
×Z
5
 
                                                            ?? ,?? =(1,0)·(0,1)=(0,0) 
?Z
5
×Z
5
 is not an I.D. 
2.4 Let ?? ={?? :?? =[?? ,?? ]?R|?? is continuous } . Show ?? is a commutative ring with 
1 under pointwise addition and multiplication. Determine whether ?? is an integral 
domain. Explain. 
(2010 : 15 Marks) 
Solution: 
Given 
?? ={?? :?? =[0,1]?R|?? is continuous } 
Let ?? ,?? ,h??? 
A. Addition: 
Page 5


Edurev123 
2. Rings 
2.1 How many proper, non-zero ideals does the ring Z
????
 have? Justify your 
answer. How many ideals does the ring Z
????
?Z
????
 have? Why? 
(???????? :?? +?? +?? +?? =???? Marks) 
Solution: 
Let (Z
?? ,?
?? ) be a cyclic group of order ?? . 
Let ?? be a subgroup of Z
?? . 
? 
?? =(?? )=???? , f.s. ?? ?Z 
We will show ?? is an ideal of Z
?? . 
Let ?? ,?? ??? . 
????????????????????????????????????????????????????????????????? ?=?? 1
?? and ?? =?? 2
?? ????????????????????????????????????????(??) 
So,  
????????????????????????????????????????????????????????????? -?? =?? 1
?? -?? 2
=(?? 1
-?? 2
)?? ??? ?????????????????????????(???? )???? 
and let ?? ??? ;?? ?Z
?? . 
Consider 
?? =?? 1
?? (?? 1
)?? ??? (???? ) 
From (i) and (ii) 
?? is an ideal of Z
?? . 
And number of subgroup of Z
?? =?? (?? ) 
??            Number of ideal of Z
?? =?? (?? ) 
??            Number of ideal of Z
12
=?? (12) 
=?? (3.4)=4 
? Number of proper non-zero ideal of Z
12
 
=4-2=2 
Using the results, "If ?? is an ideal of ring ?? and ?? is an ideal of ring ?? then ?? ×?? is a ideal 
of ?? ×?? ." 
? Number of ideals of 
Z
12
×Z
12
?=?? (12)×?? (12)
?=4×4=16
 
2.2 Show that Z[?? ] is a unique factorization domain that is not a principal ideal 
domain ( Z is the ring of integers). Is it possible to give an example of principal 
ideal domain that is not a unique factorization domain? Z[?? ] is the ring of 
polynomials in the variable ?? with integers). 
(2009 : 15 Marks) 
Solution: 
We have to show Z[?? ] is U.F.D. 
As we know, if ?? is U.F.D., then ?? (?? ) is U.F.D 
We know, every P.I.D. is U.F.D. 
?Z is P.I.D. 
?Z is U.F.D. 
?Z[?? ] is U.F.D. 
in PI.D.; ?? ?0 and non-unit is irreducible 
?<?? > is maximal. 
In Z[?? ],?? is irreducible. 
?? =?? .?? 
Here we have two possibilities, ?? is constant and ?? is of degree 1 , or ?? is constant and ?? 
is constant and ?? is of degree 1 . 
Case 1:?? is constant, ?? is of degree 1 . 
?? ?=?? (???? +?? )
?=?????? +????
 
On comparing co-efficient, 
???? =1 and ???? =0
???? =1 and ?? =0?(??? is constant )
 
??? is unit. 
Similarly in Case II ?? is unit. 
??? is irreducible. 
As ??? ????? ,2??[?? ]. But ??? ? is not maximal 
From ?
*
),??? ? is not maximal but ?? is not irreducible. 
?Z[?? ] is not P.I.D. 
Further, no it is not possible to give an example of P.I.D. that is not U.F.D. because 
every P.I.D. is U.F.D. 
2.3 How many elements does the quotient ing 
Z
?? [?? ]
(?? ?? +?? )
 have? Is it an integral 
domain? Justify your answer. 
(2009 : 15 Marks) 
Solution: 
 Let 
?? ?=
Z
5
[?? ]
??? 2
+1?
?? ?=
Z
5
[?? ]
??? 2
+1?
={?? (?? )+??? 2
+1?:?? (?? )?Z
5
(?? )}
 
By division algorithm on ?? (?? ) and ?? 2
+1 
??? (?? ) and ?? (?? ) 
?? (?? )?=?? (?? )(?? 2
+1)+?? (?? ); where ?? (?? )=0
deg??? (?? )?<deg?(?? 2
+1)
?? (?? )+??? 2
+1??=?? (?? )??? 2
+1?+?? (?? )+??? 2
+1?
?=?? (?? )??? 2
+1?+??? 2
+1?+?? (?? )+??? 2
+1?
?=??? 2
+1?+?? (?? )+??? 2
+1?
?=?? (?? )+??? 2
+1?
???? ?={?? (?? )+??? 2
+1?:?? (?? )?Z
5
[?? ]deg??? (?? )=1}
?={?? +???? +??? 2
+1?:?? ,?? ?Z
5
 
Thus, we have five choices for ' ?? ' as well as five choices for ' ?? '. 
??????????????????????????????????????????????????|?? |=5×5=25 
As ?? 2
+1 is reducible over Z
5
, because 
?? 2
+1=(?? +2)(?? +3) 
? By Chinese Remainder theorem; 
Z
5
[?? ]
??? 2
+1?
=
Z
5
[?? ]
??? +2???? +3?
?
Z
5
[?? ]
??? +2?
×
Z
5
[?? ]
??? +3?
(?g.c.d?(?? +2),(?? +3)=1) 
?Z
5
 is field and ?? +2 is irreducible polynomial over Z
5
. 
?
Z
5
[?? ]
??? +2?
 is field. 
And 
Z
5
[?? ]
??? +2?
={?? 0
+??? +2?:?? 0
?Z
5
} 
?????????????????????????????????????????|
Z
5
[?? ]
??? +2?
|=5 
????????????????????????????????????????????
Z
5
[?? ]
??? +2?
?Z
5
 
Similarly, 
Z
5
[?? ]
??? +3?
?Z
5
 
?????????????????????????????????????????????????????????????????????????
Z
5
[?? ]
??? +1?
?Z
5
×Z
5
 
?? =(1,0)?Z
5
×Z
5
 
?? =(0,1)?Z
5
×Z
5
 
                                                            ?? ,?? =(1,0)·(0,1)=(0,0) 
?Z
5
×Z
5
 is not an I.D. 
2.4 Let ?? ={?? :?? =[?? ,?? ]?R|?? is continuous } . Show ?? is a commutative ring with 
1 under pointwise addition and multiplication. Determine whether ?? is an integral 
domain. Explain. 
(2010 : 15 Marks) 
Solution: 
Given 
?? ={?? :?? =[0,1]?R|?? is continuous } 
Let ?? ,?? ,h??? 
A. Addition: 
I. Closure: (?? +?? )?? =?? (?? )+?? (?? )??? ??? ??? as ?? &?? are continuous functions. 
 ? Closure is satisfied. 
II. Associative : 
((?? +?? )+h)?? ?=?? (?? )+?? (?? )+h(?? )
?=(?? +(?? +h))?? as ?? ,?? ,h are continuous. 
 
? Associative property is satisfied. 
III. Identity : 
(?? +?? )?? =?? (?? )+?? (?? ) 
Let ?? is such that ?? (?? )=0??? ??? 
Such ?? can exist ???? , so inverse exist. 
IV. Commutative: 
(?? +?? )??ˆ =?? (?? )+?? (?? )=?? (?? )+?? (?? )=(?? +?? )?? 
??? is abelian 
So, ?? is abelian group. 
B. Multiplication: 
I. Closure : ??? (?? (?? ))??? ? Closure is satisfied. 
     II. Associative: ?? (?? (h)) follows associative law due to continuity. 
C. Distributive: ?(?? (?? +h))?? =?? (?? (?? ))+?? (h(?? ) as ?? ,?? ,h are continuous. 
? Distributive law is satisfied. 
D. Commutative : 
????????????????????????????????????????????????????? (?? ))=?? (?? (?? )) as ?? ,?? are continuous. 
So, commutative property is satisfied. 
Now, let ?? (?? )=1 be a function ??? ??? 
???? (?? )??? 
So, ?? is a commutative ring with unity. 
Now, let ?? ,?? ??? .