PlanetMath: invertibility of regularly generated ideal

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invertibility of regularly generated ideal

(Theorem)

Lemma. Let be a commutative ring containing regular elements. If $\mathfrak{a}$ , $\mathfrak{b}$ and $\mathfrak{c}$ are three ideals of such that $\mathfrak{b\!+\!c}$ , $\mathfrak{c\!+\!a}$ and $\mathfrak{a\!+\!b}$ are invertible, then also their sum ideal $\mathfrak{a\!+\!b\!+\!c}$ is invertible.

Proof. We may assume that has a unity, therefore the product of an ideal and its inverse is always . Now, the ideals $\mathfrak{b+c}$ , $\mathfrak{c+a}$ and $\mathfrak{a+b}$ have the inverses $\mathfrak{f_1}$ , $\mathfrak{f_2}$ and $\mathfrak{f_3}$ , respectively, so that

$\displaystyle \mathfrak{(b+c)f_1 = (c+a)f_2 = (a+b)f_3} = R.$

Because $\mathfrak{af_2} \subseteq R$ and $\mathfrak{cf_1} \subseteq R$ , we obtain

$\displaystyle \mathfrak{(a+b+c)(af_2f_3+cf_1f_2) = (a+b)af_2f_3+c(af_2)f_3+a(cf_1)f_2+(b+c)cf_1f_2 = af_2+cf_2 = (c+a)f_2} = R.$

Theorem 1 Let

be a commutative ring containing regular elements. If every ideal of

generated by two regular elements is invertible, then in

also every ideal generated by a finite set of regular elements is invertible.

Proof. We use induction on , the number of the regular elements of the generating set. We thus assume that every ideal of generated by regular elements ( $n \geqq 2)$ is invertible. Let $\{r_1,\,r_2,\,\ldots,\,r_{n+1}\}$ be any set of regular elements of . Denote

$\displaystyle \mathfrak{a} := (r_1),\quad \mathfrak{b} := (r_2,\,\ldots,\,r_n), \quad \mathfrak{c} := (r_{n+1}).$

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

$\displaystyle (r_1,\,r_2,\,\ldots,\,r_n,\,r_{n+1}) = \mathfrak{a+b+c}$

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

Bibliography

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

"invertibility of regularly generated ideal" is owned by pahio.

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Keywords:

invertible ideal

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Cross-references: sums, generating set, induction, finite set, ideal generated by, generated by, product, unity, sum ideal, ideals, regular elements, commutative ring
There are 3 references to this entry.

This is version 12 of invertibility of regularly generated ideal, born on 2005-04-30, modified 2008-01-14.
Object id is 6984, canonical name is InvertibilityOfRegularlyGeneratedIdeal.
Accessed 1337 times total.

Classification:

AMS MSC:	11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
	13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)

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