On Goldie Extending Modules with Condition (C2)

A module M is said to be continuous if it is extending with the condition (C₂) (cf. 6 Mohamed, S. H., Müller, B. J. (1999). Continuous and Discrete Modules. London Math. Soc. LNS, Vol. 147. Cambridge: Cambridge Univ. Press. Google Scholar]], 7 Oshiro, K. (1983). Continuous modules and quasi-continuous modules. Osaka J. Math. 20:681–694.Web of Science ®] , Google Scholar]]). In this article, we consider a 𝒢-extending module with (C₂) which is a generalization of a continuous module. First, we show that any 𝒢-extending module with (C₂) satisfies the exchange property. We also prove that, if M₁ and M₂ are 𝒢-extending modules with (C₂), then M₁ ⊕ M₂ is 𝒢-extending with (C₂) if and only if M_i is M_j-ejective (i ≠ j).