In this paper we introduce T -noncosingular modules. Rings for which all right modules are T -noncosingular are shown to be precisely those for which every simple right module is injective. Moreover, for any ring R we show that the right R-module R is T -noncosingular precisely when R has zero Jacobson radical. We also study the T -noncosingular condition in association with (strongly) FI-lifting modules.2000 Mathematics subject classification: primary 16D10; secondary 16D80.

Abstract. In this paper we introduce T -non-cosingular modules, dual Baer modules and K-modules. We prove that a module M is lifting and T -non-cosingular if and only if it is a dual Baer and K-module. Rings for which all modules are dual Baer are precisely determined. We also give a necessary condition for a finite direct sum of dual Baer modules to be dual Baer.2000 Mathematics Subject Classification. 16D10, 16D80, 16E60.

A module M is ⊕-supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕-supplemented module is not in general ⊕-supplemented. We prove that over a commutative ring R, every finitely generated ⊕-supplemented R-module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules. The structure of ⊕-supplemented modules over a commutative principal ideal ring is completely determined.2000 Mathematics Subject Classification: 16D50, 16D60, 13E05, 13E15, 16L60, 16P20, 16D80. Introduction.All rings considered in this paper will be associative with an identity element. Unless otherwise mentioned, all modules will be left unitary modules. Let R be a ring and M an R-module. Let A and P be submodules of M. The submodule P is called a supplement of A if it is minimal with respect to the property A + P = M. Any L ≤ M which is the supplement of an N ≤ M will be called a supplement submodule of M. If every submodule U of M has a supplement in M, we call M complemented. In [25, page 331], Zöschinger shows that over a discrete valuation ring R, every complemented R-module satisfies the following property (P ): every submodule has a supplement which is a direct summand. He also remarked in [25, page 333] that every module of the form M (R/a 1 ) × ··· × (R/a n ), where R is a commutative local ring and a i (1 ≤ i ≤ n) are ideals of R, satisfies (P ). In [12, page 95], Mohamed and Müller called a module ⊕-supplemented if it satisfies property (P ).On the other hand, let U and V be submodules of a module M. The submodule V is called a complement of U in M if V is maximal with respect to the property V ∩U = 0. In [17] Smith and Tercan investigate the following property which they called (C 11 ): every submodule of M has a complement which is a direct summand of M. So, it was natural to introduce a dual notion of (C 11 ) which we called (D 11 ) (see [6,7]). It turns out that modules satisfying (D 11 ) are exactly the ⊕-supplemented modules. A module M is called a completely ⊕-supplemented (see [5]) (or satisfies (D

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