Let L be a left ideal and Ka right ideal of R Let M(R) be the ideal generated by all nilpotent ideals of R (a) L LR is an ideal such that (L LR) n L n L n R for all n 1 (b) K RK is an ideal such that (K RK) n K n RK n for all n 1 (c) If L resp K is nilpotent, so is the ideal L LR resp K RK , whence...

Lets go through each part step by step a First lets show that L LR is an ideal of R Closure under addition Let x y L LR Then there exist a b L and c d R such that x a cR and y b dR Now x y a cR b dR a ...

Let L be a left ideal and Ka right ideal of R. Let M(R) be the ideal

Question:

Let L be a left ideal and Ka right ideal of R. Let M(R) be the ideal generated by all nilpotent ideals of R.

(a) L + LR is an ideal such that (L + LR)ⁿ ⊂ Lⁿ+ LⁿR for all n ≥ 1.

(b) K + RK is an ideal such that (K + RK)ⁿ ⊂ Kⁿ+ RKⁿ for all n ≥ 1.

(d) If N is a maximal nilpotent ideal of R, then R/ N has no nonzero nilpotent left or right ideals.

(e) If K [resp. L] is nil, but not nilpotent and π : R → R/ N is the canonical epimorphism, then π(K) [resp. π(L)] is a nil right [resp. left] ideal of R/ N which is not nilpotent.

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