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# Answer to Question #17267 in Abstract Algebra for Tsit Lam

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.)
1
2012-10-30T10:29:44-0400
To show that rad R is a left ideal, it suffices to check that, if a, b &isin;rad R, then r(a + b) is left quasi-regular for any r &isin; R. For any c , d&isin; R, note that c ◦ d ◦ r(a + b)= c ◦ [d + ra + rb &minus; dr(a + b)] = c◦ [d ◦ ra + (r &minus; dr)b].
Choose d such that d ◦ ra =0; then choose c such that c ◦ (r &minus; 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 &isin;rad R and s &isin; R imply as &isin;rad R. For any r &isin; R, sra is left quasiregular, so r(as) is also left quasi-regular. This shows that as &isin;rad R.
Next, we show that rad R is a quasi-regular ideal.It suffices to show that every element a &isin;rad R is left quasi-regular. Since a2 &isin; Ra is left quasi-regular, there exists b &isin; R such that b ◦ a2 = 0. But then
(b ◦ (&minus;a)) ◦ a =b ◦ ((&minus;a) ◦ a) = b ◦ a2 = 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 &isin; I, aR is right quasi-regular, therefore quasi-regular. We also see that Ra is (left) quasi-regular, so I &sube;rad R, as desired.

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