invertibility of regularly generated ideal

Lemma. Let $R$ be a commutative ring containing regular elements. If $\mathfrak{a}$ , $\mathfrak{b}$ and $\mathfrak{c}$ are three ideals of $R$ such that $\mathfrak{b\!+\!c}$ , $\mathfrak{c\!+\!a}$ and $\mathfrak{a\!+\!b}$ are invertible (http://planetmath.org/FractionalIdealOfCommutativeRing), then also their sum ideal $\mathfrak{a\!+\!b\!+\!c}$ is .

Proof. We may assume that $R$ has a unity, therefore the product of an ideal and its inverse (http://planetmath.org/FractionalIdealOfCommutativeRing) is always $R$ . Now, the ideals $\mathfrak{b+c}$ , $\mathfrak{c+a}$ and $\mathfrak{a+b}$ have the $\mathfrak{f_{1}}$ , $\mathfrak{f_{2}}$ and $\mathfrak{f_{3}}$ , respectively, so that

$\mathfrak{(b+c)f_{1}\;=\;(c+a)f_{2}\;=\;(a+b)f_{3}}\;=\;R.$

Because $\mathfrak{af_{2}}\subseteq R$ and $\mathfrak{cf_{1}}\subseteq R$ , we obtain

	$\displaystyle\mathfrak{(a+b+c)(af_{2}f_{3}+cf_{1}f_{2})}$	$\displaystyle\;=\;\mathfrak{(a+b)af_{2}f_{3}+c(af_{2})f_{3}+a(cf_{1})f_{2}+(b+% c)cf_{1}f_{2}}$
		$\displaystyle\;=\;\mathfrak{af_{2}+cf_{2}\;=\;(c+a)f_{2}}$
		$\displaystyle\;=\;R.$

Theorem. Let $R$ be a commutative ring containing regular elements. If every ideal of $R$ generated by two regular elements is , then in $R$ also every ideal generated by a finite set of regular elements is .

Proof. We use induction on $n$ , the number of the regular elements of the generating set. We thus assume that every ideal of $R$ generated by $n$ regular elements ( $n\geqq 2)$ is . Let $\{r_{1},\,r_{2},\,\ldots,\,r_{n+1}\}$ be any set of regular elements of $R$ . Denote

$\mathfrak{a}\;=:\;(r_{1}),\quad\mathfrak{b}\;=:\;(r_{2},\,\ldots,\,r_{n}),% \quad\mathfrak{c}\;=:\;(r_{n+1}).$

The sums $\mathfrak{b+c}$ , $\mathfrak{c+a}$ and $\mathfrak{a+b}$ are, by the assumptions, . Then the ideal

$(r_{1},\,r_{2},\,\ldots,\,r_{n},\,r_{n+1})\;=\;\mathfrak{a+b+c}$

is, by the lemma, , and the induction proof is complete.

References

1 R. Gilmer: Multiplicative ideal theory. Queens University Press. Kingston, Ontario (1968).

Title	invertibility of regularly generated ideal
Canonical name	InvertibilityOfRegularlyGeneratedIdeal
Date of creation	2015-05-06 15:27:47
Last modified on	2015-05-06 15:27:47
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	17
Author	pahio (2872)
Entry type	Theorem
Classification	msc 13A15
Classification	msc 11R04
Related topic	IdealMultiplicationLaws
Related topic	PruferRing
Related topic	InvertibleIdealIsFinitelyGenerated