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- 저자 : Dinesh Khurana ...et al (검색결과 1 건)
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Lifting Idempotents and Projective Covers
Dinesh Khurana ...et al KYUNGPOOK UNIVERSITY 2001 Kyungpook mathematical journal Vol.41 No.2
A necessary and sufficient condition is given so that direct summands of a module, with projective cover, have projective covers. We characterize the rings over which direct summands of cyclic (respectively finitely generated) modules, with projective covers, have projective covers. Some sufficient conditions are given so that direct summands of arbitrary modules, with projective covers, have projective covers. Let L be a left ideal of a ring R. It is said that idempotents lift modulo L in R if for every X ∈ R with x² - x ∈ L there exists an idempotent e in R such that e - x ∈ L. It is proved that if idempotents lift modulo the Jacobson radical of a ring, then idempotents also lift modulo every one sided ideal contained in the Jacobson radical. Let I be an ideal of a ring R. We prove that if idempotents lift in R modulo every left ideal contained in I then, idempotents also lift in R modulo every right ideal contained in I. On taking I = R we get that left exchange rings are right exchange.
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