generators of inverse ideal

Theorem.  Let R be a commutative ring with non-zero unity and let T be the total ring of fractionsMathworldPlanetmath of R.  If  π”ž=(a1,…,an)  is an invertible fractional idealMathworldPlanetmath ( of R with β€‰π”žβ’π”Ÿ=R,  then also the inverse ideal π”Ÿ can be generated by n elements of T.

Proof.  The equation  π”žβ’π”Ÿ=(1)  implies the existence of the elements aiβ€² of π”ž and biβ€² of π”Ÿβ€‰ (i=1,…,m) such that  a1′⁒b1β€²+β‹―+am′⁒bmβ€²=1.  Because the ai′’s are in π”ž, they may be expressed as

aiβ€²=βˆ‘j=1nri⁒jaj  (i=1,…,m),

where the ri⁒j’s are some elements of R.  Now the unity acquires the form


in which


Thus an arbitrary element b of the π”Ÿ satisfies the condition


Consequently,  π”ŸβŠ†(b1,…,bn).  Since the inverseMathworldPlanetmathPlanetmathPlanetmath inclusion is apparent, we have the equality

Title generators of inverse ideal
Canonical name GeneratorsOfInverseIdeal
Date of creation 2015-05-06 14:52:30
Last modified on 2015-05-06 14:52:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Theorem
Classification msc 13A15
Related topic FractionalIdealOfCommutativeRing
Related topic IdealGeneratedByASet
Related topic PruferRing