PlanetMath: generators of inverse ideal

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generators of inverse ideal

(Theorem)

Theorem 1 Let

be a commutative ring with non-zero unity and let

be the total ring of fractions of

. If $\mathfrak{a} = (a_1,\,\ldots,\,a_n)$ is an invertible fractional ideal of

with $\mathfrak{ab} = R$ , then also the inverse ideal $\mathfrak{b}$ can be generated by

elements of

Proof. The equation $\mathfrak{ab} = (1)$ implies the existence of the elements of $\mathfrak{a}$ and of $\mathfrak{b}$ $(i = 1, \ldots,\,m)$ such that $a_1'b_1'\!+\cdots+\!a_m'b_m' = 1$ . Because the 's are in $\mathfrak{a}$ , they may be expressed as

$\displaystyle a_i' = \sum_{j=1}^{n}r_{ij}a_j \qquad(i = 1, \ldots, m),$

where the $r_{ij}$ 's are some elements of

. Now the unity acquires the form

$\displaystyle 1 = \sum_{i=1}^{m}a_i'b_i' = \sum_{i=1}^{m}\sum_{j=1}^{n}r_{ij}a_jb_i' = \sum_{j=1}^{n}a_j\sum_{i=1}^{m}r_{ij}b_i' = \sum_{j=1}^{n}a_jb_j,$

in which

$\displaystyle b_j = \sum_{i=1}^{m}r_{ij}b_i' \,\in R\mathfrak{b} = \mathfrak{b} \qquad (j = 1, \ldots, n).$

Thus an arbitrary element

of the fractional ideal $\mathfrak{b}$ satisfies the condition

$\displaystyle b = b\!\cdot\!1 = \sum_{j=1}^{n}(a_jb)b_j \, \in Rb_1\!+\cdots+\!Rb_n = (b_1,\,\ldots,\,b_n).$

Consequently, $\mathfrak{b} \subseteq (b_1,\,\ldots,\,b_n)$ . Since the inverse inclusion is apparent, we have the equality

$\displaystyle \mathfrak{a}^{-1} = \mathfrak{b} = (b_1,\,\ldots,\,b_n).$

"generators of inverse ideal" is owned by pahio.

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Keywords:

invertible ideal, inverse ideal

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Cross-references: equality, inclusion, inverse, unity, implies, equation, proof, generated by, inverse ideal, invertible, total ring of fractions, non-zero unity, commutative ring
There are 4 references to this entry.

This is version 14 of generators of inverse ideal, born on 2004-06-20, modified 2006-10-06.
Object id is 5934, canonical name is GeneratorsOfInverseIdeal.
Accessed 1665 times total.

Classification:

AMS MSC:

13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)

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