ON RICKART MODULES

Gangyong Lee, S.Tariq Rizvi, and Cosmin S.Roman studied Rickart modules. The main purpose of this paper is to develop the properties of Rickart modules . We prove that each injective and prime module is a Rickart module. And we give characterizations of some kind of rings in term of Rickart modules.


INTRODUCTION
A module is called a Rickart module if for every , then for some .Equivalently a module is a Rickart module if and only if for every then is a direct summand of , See [1], [2] . In this paper, we give some results on the Rickart modules . In §2 ,we give characterization of the Rickart modules. Also we study the direct sum of Rickart modules. For example we prove that an -module is Rickart if and only if , for every endomorphism , see Theorem (2.2). In section 3, we give characterizations of certain classes of rings in term of the Rickart modules. For example we prove that a ring is semisimple if and only if all injective -module is Rickart , see Theorem (3.12).
Throughout this article, is a ring with identity and is a unital left -module. For a left module , will denote the endomorphism ring of . The notations mean that is a submodule, a direct summand of .

CHARACTERIZATIONS OF RICKART MODULES
In this section , we give a characterizations for the Rickart modules. Following [1] , A module is called a Rickart module if for every ), , for some . It's known that every direct summand of a Rickart module is a Rickart module. Remark 2.1: Let be an -module and be an -homomorphism.

CHARACTERIZATIONS OF RINGS BY MEANS OF RICKART MODULES
It's known The direct sum of the Rickart modules need not be a Rickart module, see [1], [2]. In this section, we give a conditions under which a direct sum of Rickart modules is a Rickart module. Proposition3.1: Let be an -module, If is -Rickart, then every cyclic submodule of is projective. In particular if is an -Rickart module, then every Principale ideal is projective ideal, i.e. , is a p.p. is isomorphic to an ideal of and hence . For the second part, since has no nilpotent elements and is an ideal in , Then is commutative[7,prop (2.1)CH1]. Thus is commutative Corollary3.3: Let be a projective indecomposable -module and has no nonzero nilpoten element. If is an -Rickart module and , then is a multiplication module. Proof: By the same argument of the proof of Cor(3.2) , is a commutative and hence is multipliction [8] Recall that an -module is called an SIP module if the intersection of any two direct summands of is also a direct summand of [9]. It is known that every Rickart module is an SIP module [1].
Before we give our next result , we need the following Theorem 3.4. [9]: Let be a Noetherian domain and let be an injective -module .If has the SIP, then either (1) is torsion free or (2) is torsion and for any two distinct indecomposable summands and of , =0 Now, we prove that Theorem 3.5: Let be a Noetherian domain and let be an injective -module, then the following are equivalent (1) is a Rickart module . (2) is torsion free .
is Rickart module, for every index set .