Associates and Irreducibles

Proposition 1 \( a,b \) are non-zero element of an integral domain \( R \). \( a \) divides \( b \) and \( b \) divides \( a \) if and only if there exist a unit \( u \) such that \( a=bu \)

Proof.

\( \Rightarrow: \)
• \( \exists r,s \in R : sa=b \) and \( rb=a \). Then, \( a=rsa \) and \( a(1-rs)=0 \). Since \( R \) is an integral domain and \( a \) is non-zero, \( 1-rs=0 \). Thus, \( r,s \) are units and \( a=rb \).
\( \Leftarrow: \)
• \( a=bu \implies b \mid a \). Since \( u \) is a unit, \( \exists t \in R : ut=1 \). \( at=b(ut)=b \implies a \mid b \).

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Proposition 2 \( a|b \iff \langle b \rangle \subseteq \langle a \rangle \)

Proof.

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Corollary 2.1 \( a,b \) are associates \( \iff \langle a \rangle = \langle b \rangle \)

Proof.

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Proposition 3 \( p \) is irreducible if and only if \( \langle p \rangle \) is maximal amongst all principal ideals that contain \( \langle p \rangle \).

Proof.

\( \Rightarrow: \)
• For any principal ideal \( \langle x \rangle \) that contains \( \langle p \rangle \) and is not the unit ideal, \( p \in \langle x \rangle \). So there exists \( r \in R : p=xr \). Since \( x \) divides \( p \) and \( p \) is irreducible, \( x \) is either an unit or an associate of \( p \). If \( x \) is an unit, then \( \langle x \rangle \) is the unit ideal. Thus, \( x \) is an associate of \( p \). Based on the previous corollary, \( \langle x \rangle = \langle p \rangle \) and \( \langle p \rangle \) is maximal amongst all principal ideals that contain \( \langle p \rangle \).
\( \Leftarrow: \)
• For any \( r \) that divides \( p \), \( \langle p \rangle \subseteq \langle r \rangle \). Since \( \langle p \rangle \) is maximal amongst all principal ideals, \( \langle p \rangle = \langle r \rangle \) or \( \langle r \rangle \) is the unit ideal. So, \( r \) is either an associate of \( p \) or an unit. Thus, \( p \) is irreducible.

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Proposition 4 In an integral domain \( R \), every prime is irreducible.

Proof. \( p \) is a prime in an integral domain \( R \). For any \( a,b \in R \) such that \( p=ab \). Since \( p \) is a prime, WLOG, let’s assume \( p|a \). Then there exist \( t \in R \) such that \( pt=abt=a \). So, \( a(bt-1)=0 \). Since \( R \) is an integral domain and \( a\neq 0 \), \( bt=1 \) and \( b \) is an unit. Therefore, \( a \) is an associate of \( p \) and \( p \) is irreducible.

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