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By L. Mirsky

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Natural is r i g h t TF, states Theorem. IN P R O J E C T I V E S . 1 in -- of of m o d - R , is t o r ~ i o n l e s s , if this. R MODULES is, Q = Q(R) for every a cogenerator of 477, the e x p o n e n t case w h e n first torsionless that asking when The the m o r e [60, p. left Noetherian, (not TORSIONLESS ( ,R) an e m b e d d i n g M --~ R n M** --+ Rn). A module M I is an e m b e d d i n g . induces R Clearly, is t o r s i o n l e s s iff the canonical map -~ R M ( if ,f(m) .... ,f , 1 n then left of h converts R Rn--~ exact exactness ---~ 0 M ~ into 0 --~ M** --~ R n exact.

Proof: Muller [24, Thm. ii] has shown that for stable FBN rings, the left localization of R at P coincides with the right localization of R at P. The result now follows from proposition i. o It is of interest to remark that if right noetherian is 50 omitted from the hypotheses localizable at every prime Example: commutative S-module, of proposition ideal. Let S = F[[x]], field F. i, R may not be left the power Let M = F[[x]] series ring over a viewed naturally but as a right F[[x]]-module as a left via the quotient ring epimorphism F[[x]] -~ F -+ o .

Let then torsion radical. and so and RR for of the equivalence S be ~ r i t a equivalent rings. Let F: classes R-Mod, which implies that has finite length. ~ is R If R has finite reduced S. The proof will make use of the characterization rank given in Theorem i. has only finitely many If the torsion radical ~i is cogenerated n has finite length with respect to ~ = ni=l~ i. ,S n. RR rank (on the left), then so does Proof. R R-Mod is maximal. ,n Then if every prime torsion radical is maximal, each simple module defines a maximal nonisomorphic The then every prime torsion radical is maximal by Conversely, only finitely many maximal by is a left Noethrian ring, then be a ring with finite reduced rank (on the left) such that only if every prime torsion radical of If R this result to certain rings with finite reduced rank.

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