# invertibility of regularly generated ideal

Lemma. β Let $R$ be a commutative ring containing regular elements. βIf $\mathfrak{a}$, $\mathfrak{b}$ 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, then also their sum idealβ $\mathfrak{a\!+\!b\!+\!c}$β is invertible.

Proof. βWe may assume that $R$ has a unity, therefore the product of an ideal and its inverse is always $R$.β Now, the idealsβ $\mathfrak{b+c}$,β $\mathfrak{c+a}$β andβ $\mathfrak{a+b}$β have the inverses $\mathfrak{f_{1}}$, $\mathfrak{f_{2}}$ 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

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

$\displaystyle\;=\;\mathfrak{af_{2}+cf_{2}\;=\;(c+a)f_{2}}$ | |||

$\displaystyle\;=\;R.$ |

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

Proof. βWe 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 invertible.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, invertible.β Then the ideal

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

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

## References

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