In this paper the structure of rings with dual continuous right ideals is discussed. The main result is the following: If R is a ring with (Jacobson) radical nil, and all of its finitely generated right ideals are dual continuous, then R -(Q J) where S is a finite direct sum of local rings each of which has its radical square zero, or is a right valuation ring, 7* is semiprimary right semihereditary ring, and M is an (5, r)-bimodule such that all of its finitely generated 7"-submodules are

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... ctive. A partial converse of this result is obtained: any matrix ring of the above type with M = 0 has all of its finitely generated right ideals dual continuous. Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.82, on 27 Jul 2018 at 05:24:46, subject to the Cambridge Core terms of THEOREM 2.1. A ring R is (semi) perfect if and only if every (finitely generated) quasi-projective R-module is d-continuous. COROLLARY 2.2. A ring R is semiperfect if and only if R R is d-continuous. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700018723 Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.82, on 27 Jul 2018 at 05:24:46, subject to the Cambridge Core terms of Saad Mohamed [6] THEOREM 3.4. Let R be a local ring with J nil. Then R is a right dcf-ring if and only if (i)/ 2 = 0, or (ii) R is a right valuation ring. PROOF. Necessity follows by the above lemma. Conversely, it is obvious that any local ring with J 2 = 0 is a right dcf-ring-in fact it has every proper right ideal semisimple. Assume that R is of type (ii). Let A be a finitely generated right ideal of R. Since R is a right valuation ring, A = aR for some element a £ R. By Lemma 2.8, r{a) is a two-sided ideal of R. Hence aR is quasi-projective by Wu and Jans (1967) . Since R is semiperfect, A is d-continuous by Theorem 2.1. This completes the proof. COROLLARY 3.5. Any local right dcf-ring with J nil is a right dc-ring whenever J ^ Rad J. PROOF. If J 1 = 0, the result is obvious. Let R be a right valuation ring with J ¥= Rad J. Let x G / -Rad / . As Rad J is a maximal submodule of / , we get J -xR. Hence R is a principal right ideal ring with descending chain condition. Hence R is a right dc-ring. By Lemma 3.3 and Theorem 3.4 we have the following: COROLLARY 3.6. Let R be a right dcf-ring with J nil. If e is an indecomposable idempotent of R, then eRe is also a right dcf-ring. Next we prove LEMMA 3.7. Let e be an indecomposable idempotent in a right dcf-ring with J nil. If eR is not an ideal, then eRe is a division ring.