Abstract. We study the geometry of r-lightlike submanifolds M of a semi-Riemannian manifoldM with a semi-symmetric non-metric connection subject to the conditions; (a) the screen distribution of M is totally geodesic in M , and (b) at least one among the r-th lightlike second fundamental forms is parallel with respect to the induced connection of M . The main result is a classification theorem for irrotational r-lightlike submanifold of a semi-Riemannian manifold of index r admitting a semisymmetric non-metric connection.

Keskin and Harmanci defined the family B(M, X) = {A ≤ M | ∃Y ≤ X, ∃f ∈ Hom R (M, X/Y), Ker f /A M/A}. And Orhan and Keskin generalized projective modules via the class B(M, X). In this note we introduce X-local summands and X-hollow modules via the class B(M, X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang's result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K ∈ B(H, X), if H ⊕ H has the internal exchange property, then H has a local endomorphism ring.

Abstract. A large number of series and integral representations for the Stieltjes constants (or generalized Euler-Mascheroni constants) γ k and the generalized Stieltjes constants γ k (a) have been investigated. Here we aim at presenting certain integral representations for the generalized Stieltjes constants γ k (a) by choosing to use four known integral representations for the generalized zeta function ζ(s, a). As a by-product, our main results are easily seen to specialize to yield those corresponding integral representations for the Stieltjes constants γ k . Some relevant connections of certain special cases of our results presented here with those in earlier works are also pointed out.