On Large-Lifting and Large-Supplemented Modules

In this paper, we introduce the concepts of Large-lifting and Large-supplemented modules as a generalization of lifting and supplemented modules. We also give some results and properties of this new kind of modules.


Introduction
Throughout this paper, we assume that R is a commutative ring with identity. A submodule N of an R-module M is called Large (essential) submodule in M, ( ) . if for every nonzero submodule K of M, then [1]. A proper submodule N of an Rmodule M is called small ( ), if for any submodule K of M such that implies that [1].Assume that N and K are submodules of M, where M is R module, then N is called supplement of K in M, if N is minimal with respect to the property . This is equivalent to and , if every submodule of M has a supplement in M, then M is called supplemented module [2]. An R-module M is called lifting, if for every submodule N of M there exists a submodule K of N such that and where H be a submodule of M ,equivalently M is called lifting, if and only if for every submodule N of M there exists a submodule K of N such that and [2]. In [3], we give the concept of Large-small (L-small) submodule , it is given as follows; Let N be a proper submodule of M , then N is called L-small submodule of M ( , if where , then K is essential submodule of M ( K . In [4], we also give the concept of Large-coessential )L-coessential ) submodule . It is given as follows; Let M be an R-module and K, N are submodules of M such that ,then K is said to be Large-coessential submodule, if . This paper consists two sections, in section one we  [2], hence L-lifting by (1). Thus as Z-module is Llifting. (5) Let M be a semisimple module, then M is lifting if and only if M is L-lifting. (6) Every hollow module is lifting [2], hence L-lifting by (1). Thus as Z-module is hollow, so it is L-lifting.
Recall that an R-module M is called L-hollow module if every proper submodule of M is L-small submodule in M [3]. , so that M is L-lifting. The converse of previous remark is not true, the following example: as Z-module is Llifting by (4) but not L-hollow by [3]. Remark 2.4: Every Local module is hollow so L-hollow [3], hence it is L-lifting by Remark(2.3), where an R-module M is called local if it is hollow and has a unique maximal submodule [5].

Large-Supplemented modules
In this section we introduce the concept of Large-supplemented modules. Some results are also given . (1) Every supplemented module is L-supplemented. Proof: Let M be a supplemented and N be a submodule of M, then N is a supplement of K in M, so and hence by [3], so N is L-supplement of K in M, hence M is L-supplemented.
(2) Next example indicates that the converse of (1) is not true. Example: Z as Z-module is L-supplemented since let , is L-supplement of since and , but Z is not supplemented since is not supplement in Z since and but is not small in , since { ̅ } is the only small submodule.    . Now ,and since M is faithful, finitely generated and multiplication, then M is cancellation by [8], so also we have , hence . To show , let H be an ideal of R such that , so and since , then so so we get the result, and hence J is Lsupplement of I in R. The characterization of L-supplement submodules is given in the next theorem. Theorem 3.9: Let M be an R-module and N, K are submodules of M, then the following statements are equivalent: 1-K is L-supplement of N in M.

2-
and for every non-essential submodule H of K, then .

Proof:
Assume K is L-supplement of N in M, so we have and and suppose where H is non-essential submodule of K, so by modular law, and since so we have and this is a contradiction, so that . From (2) , we must show . Let such that , if U is non-essential submodule of K, then by assumption , so and this is a contradiction, so that , hence , and we get K is L-supplement of N in M.