Question #17267

Show that rad R is a quasi-regular ideal which contains every quasi-regular left (resp. right) ideal of R. (In particular, rad R contains every nil left or right ideal of R.)

Expert's answer

To show that rad *R is a left ideal*, it suffices to check that, if *a, b **∈*rad *R*, then *r*(*a *+ *b*) is left quasi-regular for any *r **∈** R*. For any *c , d**∈** R*, note that *c ◦ d ◦ r*(*a *+ *b*)= *c ◦ *[*d *+ *ra *+ *rb − dr*(*a *+ *b*)] = *c◦ *[*d ◦ ra *+ (*r − dr*)*b*]*.*

Choose*d *such that *d ◦ ra *=0; then choose *c *such that *c ◦ *(*r − dr*)*b *= 0. The above equation shows that *c ◦ d *is a left quasi-inverse of *r*(*a*+ *b*), as desired. To show that rad *R is also a right ideal*,we must check that *a **∈*rad *R *and *s **∈** R *imply *as **∈*rad *R*. For any *r **∈** R*, *sra *is left quasiregular, so *r*(*as*) is also left quasi-regular. This shows that *as **∈*rad *R*.

*Next, we show that *rad *R is a quasi-regular ideal*.It suffices to show that every element *a **∈*rad *R *is left quasi-regular. Since *a*^{2} *∈** Ra *is left quasi-regular, there exists *b **∈** R *such that *b ◦ a*^{2} = 0. But then

(*b ◦ *(*−a*)) *◦ a *=*b ◦ *((*−a*) *◦ a*) = *b ◦ a*^{2} = 0

shows that*a *is left quasi-regular. By definition, rad *R *contains every quasi-regular left ideal. Let *I *be any quasi-regular *right *ideal. For *a **∈** I*, *aR *is right quasi-regular, therefore quasi-regular. We also see that *Ra *is (left) quasi-regular, so *I **⊆*rad *R*, as desired.

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