infinitude of inverses

Proposition 1.

Let R be a ring with 1.

  1. 1.

    If aR has a right inverseMathworldPlanetmathPlanetmath but no left inverses, then a has infinitely many right inverses.

  2. 2.

    If aR has more than one right inverse, then a has infinitely many right inverses.

  1. 1.

    Let ab=1. Define b0=b,b1=1-b0a+b0,,bi+1=1-bia+bi, Then, by inductionMathworldPlanetmath, we see that abi=a-abi-1a+abi-1=a-a+1=1. Next we want to show that bibj if ij. Suppose i>j and bi=bj. Again by induction, we have

    bj=bi=1+(1-a)++(1-a)i-j-1+bj(1-a)i-j (1)

    If we let c=1+(1-a)++(1-a)i-j-1 then (1-a)c=c(1-a)=(1-a)+(1-a)2++(1-a)i-j=c-1+(1-a)i-j. So Equation 3 can be rewritten as c=bj-bj(1-a)i-j=bj(1-(1-a)i-j)=bjca. Then cbj=bjcabj=bjc. Now, note that for mn, (1-a)nbjm=(1-a)n-m(bj-1)m. This implies that

    cbji-j-1 = bji-j-1+(bj-1)bji-j-2++(bj-1)i-j-1
    = g(bj)+(bj-1)i-j-1.

    On the other hand, we also have

    cbji-j-1 = bjcbji-j-2
    = bj(bji-j-2+(bj-1)i-j-3++(1-a)(bj-1)i-j-2)
    = g(bj)+bj(1-a)(bj-1)i-j-2.

    So combining the above two equations, we get (bj-1)i-j-1=bj(1-a)(bj-1)i-j-2. Let d=(bj-1)i-j-2, then (bj-1)d=bj(1-a)d=bjd-bjad. Simplify, we have d=bjad. Expanding d, then

    bji-j-2++(-1)i-j-2 = (bja)(bji-j-2++(-1)i-j-2)
    = bjabji-j-2++bja(-1)i-j-2
    = bji-j-2++(-1)i-j-2bja.

    Then 1=bja and we have reached a contradictionMathworldPlanetmathPlanetmath.

  2. 2.

    For the next part, notice that if b and c are two distinct right inverses of a, then neither one of them can be a left inverse of a, for if, say, ba=1, then c=(ba)c=b(ca)=b. So we can apply the same technique used in the previous portion of the problem. Note that if bja=1, then


    Multiply bj-1 from the right, we have


    Thus bj-1a=1. Keep going until we reach ba=1, again a contradiction.

Remark. The first part of the above propositionPlanetmathPlanetmathPlanetmath implies that a finite ring is Dedekind-finitePlanetmathPlanetmath.

Title infinitude of inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath
Canonical name InfinitudeOfInverses
Date of creation 2013-03-22 18:17:30
Last modified on 2013-03-22 18:17:30
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Theorem
Classification msc 16U99