On a Class of Dual Rickart Modules
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ON A CLASS OF DUAL RICKART MODULES R. Tribak
UDC 512.5
Let R be a ring and let ⌦R be the set of maximal right ideals of R. An R-module M is called an sd-Rickart module if, for every nonzero endomorphism f of M, Imf is a fully invariant direct summand of M. We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R-module M provided that R is a commutative Noetherian ring and Ass(M ) \ ⌦R is a finite set. In addition, we introduce and study a generalization of sd-Rickart modules.
1. Introduction Throughout the present article, all rings are associative and have an identity. All modules are unital right modules, R is a ring, ⌦R is the set of its maximal right ideals, and M is an R-module. We use J(R), EndR (M ), and E(M ) to denote the Jacobson radical of R, the endomorphism ring of M, and the injective hull of M, respectively. If R is a commutative Noetherian ring, then Ass(M ) denotes the set of all prime ideals associated with M. By Q and Z we denote the rings of rational and integer numbers, respectively. If p is a prime number, then Zp1 denotes the Pr¨ufer p-group. The notion of dual Baer modules has been recently introduced and studied in [8]. A module M is said to be dual Baer if, for every submodule N of M, there exists an idempotent e 2 S = EndR (M ) such that D(N ) = eS, where D(N ) = {' 2 S | =' ✓ N }. In [9], Lee, Rizvi, and Roman provided some motivations to study the concept of dual Rickart module, i.e., a concept related to the concept of dual Baer module. A module M is called dual Rickart (or d-Rickart) if =f is a direct summand of M for every f 2 EndR (M ). A submodule N of M is called fully invariant if f (N ) is contained in N for every R-endomorphism f of M. In [3], C˘alug˘areanu and Schultz introduced and studied the notion of stable modules. A module M is called stable if all endomorphic images are fully invariant. Clearly, if M is an R-module such that EndR (M ) is a commutative ring, then M is a stable module. Abelian groups whose endomorphism ring is commutative were characterized in [13, 14]. It is of interest to investigate the intersection C of the class of dual Rickart modules with the class of stable modules. Any element of C is called a strongly dual Rickart module. Hence, an R-module M is a strongly dual Rickart module if and only if the image of any single element of EndR (M ) is a fully invariant direct summand of M. A module M is said to have the (D2 ) condition if, for every submodule N of M for which M/N is isomorphic to a direct summand of M, N is a direct summand of M. In Section 2, we study the notion of sd-Rickart modules. It is shown that this notion differs from the notion of d-Rickart modules. Thus, it is demonstrated that the class of sd-Rickart modules is precisely the class of d-Rickart modules M for which every direct summand Centre R´egional des M´etiers de l’Education et de la Formation (CRMEF-TTH), Tanger, Morocco; e-mail: [email protected]. Published in Ukrains’kyi Matematych
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