An ideal
I is an
additive subgroup of a
ring R such that for every
i in
I and
r in
R, we have
ar in
I; see
ideal.
A principal ideal is an ideal I of a commutative ring R such that there exists an a in R such that I = {ar| r in R}; that is, I is generated by a single element a. In this case we denote the ideal as (a).
If every ideal of a given ring R is a principal ideal, we say that R is a principal ideal ring. If, further, R is an integral domain, then we call R a principal ideal domain.
Examples and Non-examples:
- In the ring of integers, Z, every ideal is principal. If we have a list of generators of an ideal I in Z, {a1, . . . , an}, then the ideal will be generated by the greatest common divisor of a1, . . . , an
- If k is a field, then k[ x ], the ring of polynomials in one variable, is a principal ideal domain. Hence, if we have an ideal generated by the polynomials {f1, . . . , fn}, then there will be a single polynomial g which generated the ideal
- If k is a field, then k[ x , y ], the ring of polynomials is not a principal ideal domain. The ideal generated by the polynomials f(x,y) = x and g(x,y) = y is not a principal ideal: since no polynomial divides both f and g except for elements of the field k, and elements of the field are not contained in the ideal, the ideal cannot be principal.