Subrings, Ideals, Quotient Rings & Euclidean Rings Quotient Rings Or Factor Rings
The concept of a quotient ring is analogous to that of quotient groups. If U is an ideal of a ring (R,+,•) then (U,+) is a normal subgroup of the commutative group (R,+).
From group theory, we know that the set R/U.= {x+U =U+ x | x ∈ R} of all cosets of U In R is a group with respect to the addition of two cosets defined by (a+U) + (b+U) = (a+b)+U for a+U,b +U ∈(R/U).
We know further that these costs are disjoint.
As addition operation is commutative left coset a+llis equal to right coset u + a In order to impose ring structure in R/U we can define multiplication of two of cosets as (a+U) (b+U) = ab +U for a+U,b +U ∈ R /U
Theorem.1 If U is an ideal of a ring R then the set R/U ={x+U|x ∈R} is a ring with respect to the induced operations of addition (+ ) and multiplication ( • ) of cosets defined : (a+ u)+(b+u)=(a+b)+U and (a+U). (b+U)=ab+U for a+Ub+u R/U.
Proof. Since (R,+) is a commutative group, the quotient group (R/U,+) is also commutative, In order to show that (R,+,•) is a ring we must show that
(1) multiplication of cosets is well defined,
(2) multiplication is associative and (3) distributive laws hold.
(1) Let \(a+U=a_1+U\) and \(b+U=b_1+U \text {. }\)
Then \(a=a_1+u_1\) and \(b=b_1+u_2\) for \(u_1, u_2 \in U \text {. }\)
ab = \(\left(a_1+u_1\right)\left(b_1+u_2\right)=a_1 b_1+a_1 u_2+u_1 b_1+u_1 u_2\)
Since U is an ideal, \(a_1 u_2, u_1 b_1, u_1 u_2 \in U\)
∴ \(a b-a_1 b_1 \in U\) and hence \(a b+U=a_1 b_1+U \Rightarrow(a+U) \cdot(b+U)=\left(a_1+U\right) \cdot\left(b_1+U\right)\)
Therefore multiplication of cosets is well defined.
Let a+U,b+U,c+U ∈R/U
(2) [(a+U).(b+U)] .(c+U) = (ab+U).(c+U) = (ab) c+U = a'(bc) +U (v a,b,c∈R) = (a +U). (bc+U) = (a+U) . [(b+U).(c+U)]
(3) (a+U). [(b+U)+(c+U)] = (a+U).[(b+ c)+U] = a(b+ c)+U = (ab+ ac)+U (v a,b,c∈R)
(ab+U)(ac+U) = (a+U) .(b+U)+(a+U). (c+U)
Similarly we can prove that [(b+U)+ (c+ U)].(a+U) = (b+U).(a+U)+(c+U).(a+U) Hence (R/U,+,•) is a ring
Definition Of Addition And Multiplication Cosets
Definition. Let R be a ring and U be an ideal of R. Then the set R/U = {x+U\x∈R] with respect to induced operations of addition and multiplication of cosets defined by (a+U)+(b+U)-(a+b) +U;(a+U). (b+U) = ab +Ufor a+U,b+U eR/U is a ring.
This ring (R/U,+,•)is called the quotient ring or factor ring, or residue class ring of R modulo U.
Note.
- It is convenient, sometimes, to denote coset a+U in R/U by the symbol a or.[a]. Then we write the sum and product of two cosets as [a]+[b] =[a+b] and [a].[b] = [ab].
- o+U = U is the zero element in the ring R/U.
- Every ring R has two improper ideals, namely, the trivial ideal {0} and the ideal R. The quotient ring of the ideal {0} is R/{0} or R /(0) = {x+ (0)| x∈ R}The quotient ring of the ideal R is R / R or Rt(l) = {x+(l) | x ∈ R}
- (a+U)+ (b+U) = (a+b)+U-,(a+U)(b+U) = ab+U
- (a+U)² =(a+U)(a+U) = a²+U
- a+U = b+U ⇔(a-b)∈U.
- a +U =U ⇔ a ∈ U
Theorem. 2. If R/U is the quotient ring prove that (1) R/U is commutative if R is commutative and (2) R/u has a unity element if R has a unity element.
Proof. (2) R is commutative => ab = ba∀ a,b ∈ R.
Let a+U,b+U ∈ R/U. (a+U) (b+U) = ab+U = ba+U =(b+U).(a+U)
∴ R/U is commutative.
(2) R has unity element => there exists 1 ∈ R so that a1 = 1a- a∀ a∈R.
Let a+U ∈ R/U. For 1∈R we have 1+U ∈R/U
We now prove that 1 +U is the unity element.
(a+U)(l+U) = a1+U = a+U and (X+U)(a+U)*=la+U = a+U V a+U eR/U
1+U is the unity element in R/U.
Note. In the quotient ring R/U, the unity element = 1 +U.
Rings Examples Of Addition And Multiplication Cosets
Example. 1 Consider Z6 = {0,1, 2,3,4,5}, the ring of integers modulo 6.
U= {0,3} is an ideal of Z6. The costs of Uin R are as follows :
0+U = {0 + 0, 0+ 3} = {0,3};l+t/ = {1 + 0,1+ 3} = {1,4}
2+U = {2 +02 + 3} = {2,5};3+t/ = {3+0, 3+ 3} = {3,0} =0+ 17
4+U = {4+ 0, 4+ 3} = {4, 1} – 1 +U and 5+U {5 + 0, 5+3} = {5, 2} = 2+U
(Z6/U) = {0+U,l+U,2+U} is the quotientring.
Note. We observe that two cosets are identical or disjoint and the union of all cosets = Z6.
Example 2. For the ring Z of all integers, we know that nZ = {nx| x∈ Z} for any n∈ Z is an additive subgroup of Z.
Let m ∈ nZ and r ∈ Z Then m = na where a∈ Z.
mr = (na)r = n (ar) and rm =r (na) = n (ar)
so that mr = rm = n (ar) = nb∈nZ where b = ar∈Z Thus nZ is an ideal of Z.
The set of all cosets of nZ in Z, namely, (Z/nZ) = {x+nZ |x∈ Z} forms a ring under the induced operations of addition and multiplication.