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**, {`a`_{1}, . . . , a_{n}}, then the ideal will be generated by the greatest common divisor of `a`_{1}, . . . , a_{n}
- 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 {`f`_{1}, . . . , f_{n}}, 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.