DUAL OJECTIVITY OF QUASI-DISCRETE MODULES AND LIFTING MODULES

Abstract. In [3], K.Oshiro and his students introduced “ojectivity (generalized injec-tivity)”, a new concept of relative injectivity, and using this injectivity we obtained some results for direct sums of extending modules. Afterward, S.H.Mohamed and B.J.Müller [9] defined a dual concept of ojectivity as follows: Definition. M is said to be N-dual ojective (or generalized N-projective) if, for any epimorphism g: N → X and any homomorphism f:M → X, there exist decompositions N = N1 ⊕ N2, M = M1 ⊕M2, a homomorphism h1: M1 → N1 and an epimorphism h2: N2 →M2, such that gh1 = f |M1 and fh2 = g|N2. The concept of relative dual ojectivity is a generalization of relative projectivity and this projectivity has an important meaning for the study of direct sums of lifting modules (cf. [6], [9]). In this paper we introduce some results on “dual ojectivity ” and apply it to direct sums of quasi-discrete modules. 1