# 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:

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

Let $x, y \in J$.

By definition of linear combination:

 $\ds x$ $=$ $\ds \sum_{i \mathop = 1}^n r_i \circ a_i$ for some $n \in \N$ and for some $r_i \in D$ where $i \in \set {1, 2, \dotsc, n}$ $\ds$ $=$ $\ds r_1 \circ a_1 + r_2 \circ a_2 + \dotsb + r_n \circ a_n$ for some $r_1, r_2, \dotsc, r_n \in D$

and:

 $\ds y$ $=$ $\ds \sum_{i \mathop = 1}^n s_i \circ a_i$ for some $n \in \N$ and for some $s_i \in D$ where $i \in \set {1, 2, \dotsc, n}$ $\ds$ $=$ $\ds s_1 \circ a_1 + s_2 \circ a_2 + \dotsb + s_n \circ a_n$ for some $s_1, s_2, \dotsc, s_n \in R$ $\ds \leadsto \ \$ $\ds -y$ $=$ $\ds -\sum_{i \mathop = 1}^n s_i \circ a_i$ $\ds$ $=$ $\ds -1_D \times \sum_{i \mathop = 1}^n s_i \circ a_i$ Product with Ring Negative: Corollary $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {-1_D} \times \paren {s_i \circ a_i}$ Ring Axiom $\text D$: Distributivity of Product over Addition $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {-\paren {s_i \circ a_i} }$ Product with Ring Negative: Corollary $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n s_i \circ \paren {-a_i}$ Product with Ring Negative

Thus:

 $\ds x + \paren {-y}$ $=$ $\ds \sum_{i \mathop = 1}^n r_i \circ a_i + \sum_{i \mathop = 1}^n s_i \circ \paren {-a_i}$ $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {r_i \circ a_i + s_i \circ \paren {-a_i} }$ $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {r_i \circ a_i + \paren {-\paren {s_i \circ a_i} } }$ $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {r_i \circ a_i + \paren {-s_i} \circ a_i}$ $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {r_i + \paren {-s_i} } \circ a_i$ Ring Axiom $\text D$: Distributivity of Product over Addition $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n t_i \circ a_i$ where $t_i - r_1 + \paren {-s_i}$ $\ds$ $\in$ $\ds J$ as $t_i \in D$

Then we have:

 $\ds x \circ y$ $=$ $\ds \paren {\sum_{i \mathop = 1}^n r_i \circ a_i} \circ \paren {\sum_{i \mathop = 1}^n s_i \circ a_i }$ $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {t_i \circ a_i}$ where $t_i \in D$ for $i \in \set {1, 2, \dotsc, n}$ $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \paren {a_i \circ t_i}$ as $\circ$ is commutative in an integral domain

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 $x \in D$.

$\blacksquare$