Set of Linear Combinations of Finite Set of Elements of Principal Ideal Domain is Principal Ideal

Theorem
Let $\struct {D, +, \circ}$ be a principal ideal domain.

Let $a_1, a_2, \dotsc, a_n$ be non-zero elements of $D$.

Let $J$ be the set of all linear combinations in $D$ of $\set {a_1, a_2, \dotsc, a_n}$

Then for some $x \in D$:
 * $J = \ideal x$

where $\ideal x$ denotes the principal ideal generated by $x$.

Proof
Let the unity of $D$ be $1_D$.

By definition of principal ideal:


 * $\ideal a = \displaystyle \set {\sum_{i \mathop = 1}^n r_i \circ a \circ s_i: n \in \N, r_i, s_i \in D}$

Let $x, y \in J$.

By definition of linear combination:

and:

Thus:

Then we have:

Thus by the Test for Ideal, $J$ is an ideal of $D$.

As $D$ is a principal ideal domain, it follows that $J$ is a principal ideal.

Thus by definition of principal ideal:


 * $J = \ideal x$

for some $x \in D$.