# 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

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

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 InvertibilityOfRegularlyGeneratedIdeal 2015-05-06 15:27:47 2015-05-06 15:27:47 pahio (2872) pahio (2872) 17 pahio (2872) Theorem msc 13A15 msc 11R04 IdealMultiplicationLaws PruferRing InvertibleIdealIsFinitelyGenerated