# invertibility of regularly generated ideal

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

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

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

Becauseβ $\mathfrak{af_{2}}\subseteq R$β andβ $\mathfrak{cf_{1}}\subseteq R$,β we obtain

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

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

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

The sums β$\mathfrak{b+c}$, β$\mathfrak{c+a}$β andβ $\mathfrak{a+b}$β are, by the assumptions, .β 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. β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