Subrings, Ideals, Quotient Rings & Euclidean Rings Principal Ideal
If a commutative ring with unity from Theorem (3) Art! 2.2 we observed that for a given a ∈ R the set {ra | r ∈ #} is an ideal in R that contains the element ‘a’.
Definition. Let R be a commutative ring with unity and a ∈ R The ideal {ra |r ∈ R) of all multiples of ‘a’ is called the principal ideal generated by ‘a’ and is denoted by (a) or (a).
An ideal Uofthe ring R is a principal ideal =>U = (ci) = {ra ) r ∈ It} for some a∈R
example. 1 The null ideal or trivial ideal {0}’ofaring# is the principal ideal generated by the zero element of R. That is null ideal = (0).
example. 2. The unit ideal or improper ideal R of a ring R is the principal ideal generated by the unity element T of the ring R. That is R =(l).
example.3. Z is a commutative ring with a unity element. The principal ideal generated by 2 ∈ Z = (2) = [2n| n ∈ Z} = the set of all even integers.
example. 4. A field has only null ideal = (0) and unit ideal =F =(l) which are principal ideals.
Definition. (Principal ideal ring). A commutative ring R with unity is a principal ‘ ideal ring if every ideal in R is a principal ideal
D is a principal ideal domain => every ideal U in D is in the form U (a) for some a ∈D.
Principal Ideal Theorem In Euclidean Rings Explained
Theorem. 1. A field is a principal ideal ring
Proof. A field F has only two ideals, namely, U = (0) and U = F = {l).
But U = (0) and U = (l) are principal ideals. the field F is a principal ideal ring.
Theorem. 2. ring of integers the principal ideal ring, (or) Every ideal Z is of Z is a principal ideal.
Proof. Let U be the ideal of Z and U = {0}. Then U is generated by the zero element.
.’. U = (0) is a principal ideal. Let U be an ideal of Z and U≠ (0).
there exists a ∈U so that a ≠ 0,
Since u ⊂ Z, one of a, -a must be a positive integer.
∴ the set of positive integers U+ in U is non-empty. by the well-ordering principle, U+ has the least member, say, b.
We now prove that U = {b) = the principal ideal generated by ‘b’.
Let x∈U. Since x,b are integers and b≠0 there exist q,r ∈ Z
such that x = bq+r; 0 <,r<b (Division algorithm).
b ∈ U,q ∈ Z and u is an ideal => bq ∈ U. x∈U,bq ∈U => x-bq =r ∈U .
Now r ∈U 0 <r < b and b is the least member in U+ =>r = 0.
x-bq =r=>x-bq = 0 => x = bq .
Hence x∈U=>x=bq for q∈Z =>U = (b).
∴ every ideal U of Z is a principal ideal. Hence Z is a principal ideal ring.
Examples Of Principal Ideal Theorem In Subrings And Quotient Rings
Note. 1. Principal ideal rings that are also integral domains, such as rings of integers Z are called principal ideal domains
2. If Z is the ring of integers then the principal ideal generated by a∈ Z is ] {aq \q∈Z} = (a).