Math. Sci. - downloads.hindawi.com618 Z. JULE AND D. XIANNENG THEOREM2.1.If Ris a semiprime...
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Internat. J. Math. & Math. Sci. VOL. 17 NO. 3 (1994) 617-618 617 SEMIPRIME SF-RINGS WHOSE ESSENTIAL LEFT IDEALS ARE TWO-SIDED ZHANG JULE and DU XHIANNENG Anhui Normal University Wuhu 241000, P.R. of China (Received April 21, 1992) ABSTRACT. It is proved that if R is a semiprime ELT-ring and every simple right R-module is flat then R is regular. Is R regular if R is a semiprime ELT-ring and every simple right R-module is flat? In this note, we give a positive answer to the question. KEY WORDS AND PHRASES. (Von Neumann) regular ring, SF-ring, ELT-ring. 1991 AMS SUBJECT CLASSIFICATION CODE. 16A30. 1. INTRODUCTION. In [1] Yue Chi Ming proposed the following question: Is R regular if R is a semiprime ELT- ring and every simple right R-module is flat? In this note, we give a positive answer to the question. All rings considered in this paper are associative with identity, and all modules are unital. A ring R is (Von Neumann) regular provided that for every a R there exists R such that a aba (see [2]). R is called a strongly regular ring if for each a R,a a2R. Following [1], call R and ELT-ring if every essential left ideal is an ideal of R. We call R a right SF-ring if every simple right R-module is flat (see [31). 2. MAIN RESULTS. We begin by stating following lemmas which will be used in proof of our main result. LEMMA 1. ([4], p.30, Exercise 19) If R is a semiprime ring, then Soc(RR Soc(RR). LEMMA 2. ([5], Corollary 8.5) If R is a semiprime ring, then.every minimal left (right) ideal is generated by an idempotent. LEMMA 3. ([3], Proposition 3.2) Let R be a left (right) SF-ring. If I is an ideal of R, then R/I also is a left (right) SF-ring. LEMMA 4. ([3], Theorem 4.10) Let R be a left (right) SF-ring. If every maximal right (left) ideal of R is an ideal, then R strongly regular. LEMMA 5. If R is a semiprime ELT and right SF-ring, then R is fully left (right) idempotent. PROOF. From Lemma 1, Soc(RR)=Soc(RR). Now we write instead of Soc(nR). By Lemma 2, S is fully left (right) idempotent. Since R is an ELT-ring, and every maximal left ideal of R/S is an image of a maximal essential left ideal of R under the natural map v:R--R/S, hence every maximal left ideal of R/S is an ideal. By Lemma 3, R/S is a right SF-ring. It follows from Lemma 4 that R/S is strongly regular, whence R/S is fully left (right) idempotent. Since S is fully left (right) idempotent, then R is fully left (right) idempotent. Now we prove our main result which gives a positive answer to the question raised in [1].
Transcript of Math. Sci. - downloads.hindawi.com618 Z. JULE AND D. XIANNENG THEOREM2.1.If Ris a semiprime...
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Internat. J. Math. & Math. Sci.VOL. 17 NO. 3 (1994) 617-618
617
SEMIPRIME SF-RINGS WHOSE ESSENTIAL LEFT IDEALS ARE TWO-SIDED
ZHANG JULE and DU XHIANNENGAnhui Normal University
Wuhu 241000, P.R. of China
(Received April 21, 1992)
ABSTRACT. It is proved that if R is a semiprime ELT-ring and every simple right R-module isflat then R is regular. Is R regular if R is a semiprime ELT-ring and every simple right R-moduleis flat? In this note, we give a positive answer to the question.
KEY WORDS AND PHRASES. (Von Neumann) regular ring, SF-ring, ELT-ring.1991 AMS SUBJECT CLASSIFICATION CODE. 16A30.
1. INTRODUCTION.In [1] Yue Chi Ming proposed the following question: Is R regular if R is a semiprime ELT-
ring and every simple right R-module is flat? In this note, we give a positive answer to thequestion.
All rings considered in this paper are associative with identity, and all modules are unital.
A ring R is (Von Neumann) regular provided that for every a R there exists R such thata aba (see [2]). R is called a strongly regular ring if for each a R,a a2R. Following [1], call Rand ELT-ring if every essential left ideal is an ideal of R. We call R a right SF-ring if everysimple right R-module is flat (see [31).2. MAIN RESULTS.
We begin by stating following lemmas which will be used in proof of our main result.
LEMMA 1. ([4], p.30, Exercise 19) If R is a semiprime ring, then Soc(RR Soc(RR).LEMMA 2. ([5], Corollary 8.5) If R is a semiprime ring, then.every minimal left (right)
ideal is generated by an idempotent.LEMMA 3. ([3], Proposition 3.2) Let R be a left (right) SF-ring. If I is an ideal of R, then
R/I also is a left (right) SF-ring.LEMMA 4. ([3], Theorem 4.10) Let R be a left (right) SF-ring. If every maximal right
(left) ideal of R is an ideal, then R strongly regular.LEMMA 5. If R is a semiprime ELT and right SF-ring, then R is fully left (right)
idempotent.
PROOF. From Lemma 1, Soc(RR)=Soc(RR). Now we write instead of Soc(nR). ByLemma 2, S is fully left (right) idempotent. Since R is an ELT-ring, and every maximal leftideal of R/S is an image of a maximal essential left ideal of R under the natural map v:R--R/S,hence every maximal left ideal of R/S is an ideal. By Lemma 3, R/S is a right SF-ring. Itfollows from Lemma 4 that R/S is strongly regular, whence R/S is fully left (right) idempotent.Since S is fully left (right) idempotent, then R is fully left (right) idempotent.
Now we prove our main result which gives a positive answer to the question raised in [1].
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618 Z. JULE AND D. XIANNENG
THEOREM 2.1. If R is a semiprime ELT and right SF-ring, then R is regular.PROOF. From Lemma 5, R is a fully left (right) idempotent ring. If P is a prime ideal of
R, then it is easy to know that RIP is a fully right idempotent ring. Since R is ELT, this impliesthat R/P is an ELT-ring. By (see [6], Corollary 6), R/P is regular. Considering that R is fullyidempotent, thus R is a regular ring (see [21, Corollary 1.18).
ACKNOWLEDGEMENT. This research was supported by Anhui Education Council of China.
REFERENCES
1. YUE CHI MING, R., On Von Neumann regular rings V, Math. J. Okayama Univ. 22(1980), 151-160.
2. GOODEARL, K.R., Von Neumann Regular Rings, Pitman, London, 1979.3. REGE, M.B., On Von Neumann regular rings and SF-rings, Math. Japonica 31 No. 6
(1986), 927-936.4. GOODEARL, K.R., Ring Theory: Non-singular Rings and Modules, Dekkar, New York,
1974.
5. FAITH, C., Algebra I: Rings, Modules, and Categories, Springer-Verlag, 1981.6. HIRANO, Y. & TOMINAGA, H., Regular rings, V-rings and their generalizations,
Hiroshima Math. J. 9 (1979), 137-149.
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