New generalizations of lifting modules


In this paper, we call a module M almost \mathcal{I}-lifting if, for any element \phi\in S=End_R(M), there exists a decomposition r_M\ell_S(\phi)=A\oplus B such that A\subseteq \phi M and \phi M\cap B\ll M. This definition generalizes the lifting modules and left generalized semiregular rings. Some properties of these modules are investigated. We show that if f_1+\cdots + f_n=1 in S, where f_i {^{ ,}}s are orthogonal central idempotents, then M is an almost \mathcal{I}-lifting module if and only if each f_iM is almost \mathcal{I}-lifting. In addition, we call a module M \pi-\mathcal{I}-lifting if, for any \phi\in S, there exists a decomposition \phi^nM=eM\oplus N for some positive integer n such that e^2=e\in S and N\ll M. We characterize semi-\pi-regular rings in terms of \pi-\mathcal{I}-lifting modules. Moreover, we show that if M_1 and M_2 are abelian \pi-\mathcal{I}-lifting modules with Hom_R(M_i, M_j)=0 for i\neq j, then M=M_1\oplus M_2 is a \pi-\mathcal{I}-lifting module.

DOI Code: 10.1285/i15900932v36n2p49

Keywords: Lifting module; $\mathcal{I}$-Lifting module; Semiregular ring; Semi-$\pi$-regular ring

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