Divisors of Elements in Commutative Rings

So far we have discussed the term "divisor" with regards to integers and polynomials over a field $F$. We will now extend the notion of a divisor to a general commutative ring.

Definition: Let $(R, +, \cdot)$ be a commutative ring and let $a, b \in R$. Then $b$ is said to be a Divisor of $a$ denoted $b | a$ if there exists an element $q \in R$ such that $a = bq$.

Many of the proofs regarding divisors of integers and polynomials also applies to divisors of elements in a commutative ring.

Theorem 1: Let $(R, +, \cdot)$ be a commutative ring. Then: a) If $a | b$ and $b | c$ then $a | c$. b) If $a | b$ then $a | bc$. c) If $a | b$ and $a | c$ then for all $x, y \in R$, $a | (bx + cy)$.

• Proof of a) Since $a | b$ and $b | c$ there exists $q_1, q_2 \in R$ such that $aq_1 = b$ and $bq_2 = c$. Plugging the second equation into the first yields:

(1)

\begin{align} \quad a(q_1q_2) = c \end{align}

• So $a | c$. $\blacksquare$

• Proof of b) Since $a | b$ there exists a $q \in R$ such that $aq = b$. Multiply both sides of this equation by $c$ to get $a(cq) = bc$. So $a | bc$. $\blacksquare$

• Proof of c) Since $a | b$ and $a | c$ there exists $q_1, q_2 \in R$ such that $aq_1 = b$ and $aq_2 = c$. So for any $x, y \in R$ we have that $a(q_1x) = bx$ and $a(q_2y) = cy$. Adding these equations yields:

(2)

\begin{align} \quad a(q_1x + q_2y) = bx + cy \end{align}

• So $a | (bx + cy)$. $\blacksquare$

Theorem 2: Let $(R, +, \cdot)$ be a commutative ring and let $a, b \in R$. Then $aR \subseteq bR$ if and only if $b | a$.

• Proof: $\Rightarrow$ Suppose that $aR \subseteq bR$. Then since $a \in aR$ we have that $a \in bR$. Since $bR = \{ bq : q \in R \}$ we see that $a = bq$ for some $q \in R$. So $b | a$.

• $\Leftarrow$ Suppose that $b | a$. Then there exists an element $q \in R$ such that $a = bq$.

• Let $x \in aR$. Then $x = ay$ for some $y \in R$. Thus $x = b(qy)$ which shows that $x \in bR$. Therefore $aR \subseteq bR$. $\blacksquare$