Let R be a commutative ring with unity and a in R. Then <<a>>={ra:r in R}=Ra=aR

Khaleesi Herbert 2021-03-02 Answered
Let R be a commutative ring with unity and a in R. Then a={ra:rR}=Ra=aR
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Expert Answer

toroztatG
Answered 2021-03-03 Author has 98 answers

Let consider,
0R,0.a=0Ra,
RaφandRaR
Let x,yRa then x=r1a,y=r2a where r1,r2R
Now (xy)=(r1r2)a=ra where r=r1r2R,xyRa(1)
Let xRa,rR
x.r=(r1a)(r)=(r1r)a=ra where r1r=rR
Since R is commutative x.r=r.x(2)
xRa,rRx.r=r.xRa
From (1) and (2) Ra is an ideal of R.
Since R is commutative then aR=Ra
Therefore a={ra,rR}=Ra=aR where aR,rR

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