By Eric Jespers

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Every principal left ideal is a direct summand of RR . 4. Every finitely generated left ideal is generated by an idempotent. 5. Every finitely generated left ideal is a direct summand of RR . A ring that satisfies these condition is called Von Neumann regular (J. Von Neumann, 1903 - 1957) . Proof. (1) implies (2). Let a ∈ R and consider the principal left ideal Ra. Let x ∈ R so that axa = a. Then e = xa is an idempotent. Indeed, e = xa = xaxa = e2 . Clearly Re ⊆ Ra. As a = ae it follows that Ra = Re.

Since B(R) is a semiprime ideal we get that a ∈ B(R), a contradiction. Let N be a maximal nilpotent ideal in R. Then N ⊆ B(R) and R/N does not have nonzero nilpotent ideals. So R/N is semiprime and N = B(R) is nilpotent. By the first part of the proof, every nil left ideal contained in B(R), and thus it is nilpotent. If aR is nil then so is Ra. Hence Ra is nilpotent. It follows that aR is nilpotent. This proves the result. 11 If R is a left Noetherian ring, then the prime radical contains all nil left and right ideals.

1. 2 that r ∈ J(R). 4 Let R be a ring. 1. J(R) is the largest left (and two-sided) ideal L so that 1 + L ⊆ U (R) (the group of invertible elements of R). 2. J(R) is the intersection of all maximal right ideals. 5 Let R be a ring and I an ideal contained in J(R). 1. J(R/I) = J(R)/I, 2. J(R/J(R)) = {0}. f Because I ⊆ J(R) we get that I is contained in every maximal left ideal of R. Moreover, the maximal left ideals of R/I are the sets L/I with L a maximal left ideal of R that contains I. Hence (1) follows.

### Associative algebra by Eric Jespers

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