Studia Scientiarium Mathematicarum Hungarica 28. (1993)

1-2. szám - Conchigdorzh R.: Generalized p.p. rings and rings of pi-regular quotients

Studio Scientiarum Mathematicarum Hungarica 28 (1993), 1-15 GENERALIZED P.P. RINGS AND RINGS OF tt-REGULAR QUOTIENTS R. GONCHIGDORZH The recent papers [8], [10] and [6] have been devoted to the study of non­­commutative and commutative generalized p.p. rings with identity. In the noncommutative case the existence of classical right rings of quotients was assumed. In the present paper we continue these investigations for noncom­mutative normal generalized p.p. rings not necessarily having an identity element. In this case we cannot assume existence of the classical ring of quotients, because it may happen that such a ring has no cancellative ele­ment. So, it is a natural question to find a construction of ring-extensions for generalized p.p. normal rings not necessarily having identity which can be considered as a substitution of the construction of the classical rings of quo­tients. We shall present a construction of a ring of right 7r-regular quotients of a normal generalized p.p. ring. This is a generalization of the rings of right regular quotients of reduced rings introduced by the author in [4], The latter one was used in [5] for characterizing semihereditary and hereditary reduced rings without assuming the existence of an identity element (similar results for semihereditary and hereditary rings with identity were obtained by M. Ohori in [10]). The first section is preliminaries and there we recall some definitions and results needed later. In Section 2 we shall define rings of right 7T-regular quotients of normal generalized p.p. rings and we shall give some sufficient and necessary con­ditions for the existence of rings of right 7r-regular quotients and corollaries for rings with identity. These results can be considered as characterizations of normal generalized p.p. rings with rings of right 7r-regular quotients. So, there is some generalization of results in [8]. In the third section we shall consider normal generalized p.p. rings with Köthe radical and there a characterization will be obtained for a normal generalized p.p. ring having a ring of right 7T-regular quotients with Pierce stalks which are local rings. The last section is devoted to a characterization of normal generalized p.p. rings with 7r-regular rings of right 7r-regular quotients. This is a gener­alization of some results in [10]. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A08; Secondary 16A30. Key words and phrases. Generalized p.p. ring, normal ring, annihilator, Pierce sheave, the ring of rr-regular quotients, Ore ring. Akadémiai Kiadó, Budapest MAGYAR TUD0MÁNY06 AKADSM* könyvtara

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