This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. Contemporary Mathematics Volume 609, 2014 http://dx.doi.org/10.1090/conm/609/12091 On a Class of ⊕-Supplemented Modules Burcu Ungor, Sait Halicioglu, and Abdullah Harmancı This paper is dedicated to Professor T. Y. Lam on his 70th birthday Abstract. In this paper, we introduce principally ⊕-supplemented modules as a generalization of ⊕-supplemented modules and principally lifting modules. This class of modules is a strengthening of principally supplemented modules. We show that the class of principally ⊕-supplemented modules lies between classes of ⊕-supplemented modules and principally supplemented modules. We prove that some results of ⊕-supplemented modules and principally lifting modules can be extended to principally ⊕-supplemented modules for this gen- eral setting. We obtain some characterizations of principally semiperfect rings and von Neumann regular rings by using principally ⊕-supplemented modules. Introduction Throughout this paper R denotes a ring with identity, modules are unital right R-modules. Let M be a module and N , K be submodules of M . We call K a supplement of N in M if M = K + N and K ∩ N is small in K. A module M is called supplemented if every submodule of M has a supplement in M . A module M is called lifting if for all N ≤ M , there exists a decomposition M = A ⊕ B such that A ≤ N and N ∩ B is small in M . Supplemented and lifting modules have been discussed by several authors (see [CLVW], [MM]) and these modules are useful in characterizing semiperfect and right perfect rings (see [KY], [MM], [Wi]). A module M is defined to be principally supplemented [AH], if for all cyclic submodule N of M there exists a submodule X of M such that M = N + X with N ∩ X is small in X, and a module M is called principally lifting [KY], if for all cyclic submodule N of M there exists a decomposition M = A ⊕ B such that A ≤ N and N ∩ B is small in B. Principally lifting modules are considered as generalizations of lifting modules. Following [MM], a module M is said to be ⊕- supplemented if every submodule of M has a supplement which is a direct summand of M . 2010 Mathematics Subject Classification. 13C10, 16D10, 16E50. Key words and phrases. Principally lifting module, principally supplemented module, ⊕- supplemented module, principally ⊕-supplemented module, principally semisimple module. The first author was supported by the Scientific and Technological Research Council of Turkey (TUBITAK). c 2014 American Mathematical Society 123