View source for Commutative Ring Theory/Divisibility and principal ideals

{{definition|principal ideal|Let <math>R</math> be a commutative ring. A '''principal ideal''' is a [[General Ring Theory/Principal ideals#left principal ideal|left principal ideal]] of <math>R</math>. Equivalently, it is a right principal ideal or a two-sided principal ideal of <math>R</math>.}}

{{proposition|characterisation of divisibility by principal ideals|Let <math>R</math> be a commutative ring, and let <math>a, b \in R</math>. Then <math>a|b \Leftrightarrow \langle a \rangle \ge \langle b \rangle</math>.}}

{{proof|Both assertions are equivalent to the existence of a <math>c \in R</math> such that <math>b = ac</math>.}}

{{definition|similarity|Let <math>R</math> be a commutative ring. Two elements <math>a, b \in R</math> are called '''similar''' if and only if there exists a unit <math>u \in R</math> such that <math>a = ub</math>.}}

{{proposition|similarity is an equivalence relation|Given a ring <math>R</math>, the relation of similarity defines an equivalence relation on the elements of <math>R</math>.}}

{{proof|For reflexivity, use the identity, and for symmetry, use the inverse. Suppose that <math>a = ub</math> and <math>b = vc</math>, where <math>u,v \in R^\times</math>. Then <math>a = uvc</math>, where of course <math>uv \in R^\times</math>.}}

{{proposition|in an integral domain, the generating element of a principal ideal is unique up to similarity|Let <math>R</math> be an integral domain, and let <math>\langle a \rangle \le R</math> be a principal ideal of <math>R</math>. Then if <math>\langle a \rangle = \langle b \rangle</math> for some element <math>b \in R</math>, we have <math>a = ub</math> for some <math>u \in R^\times</math>.}}

{{proof|The equation <math>\langle a \rangle = \langle b \rangle</math> implies that <math>a = xb</math> and <math>b = ya</math> for certain <math>x, y \in R</math>. Hence, <math>a = xya</math>. By cancellation (which is applicable because <math>R</math> is an integral domain), <math>xy = 1</math> and hence <math>x</math> is a unit, so that <math>a</math> and <math>b</math> are similar.}}

{{BookCat}}