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

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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


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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 $x \in D$.

$\blacksquare$


Sources

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