We call a module MR essentially retractable if HomR (M,N)¹0 for all essential submodules N of M. For a right FBN ring R, it is shown that: (i) A nonzero module R M is retractable (in the sense that HomR (M,N) ¹0 for all nonzero N£MR) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii) A non-zero module MR is essentially retractable if and only if there exists a prime ideal PÎAss (MR) such that HomR (M,N)¹0. Over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with nonzero dual. Moreover, over certain rings R, including right FBN rings, it is shown that a nonsingular module M with enough uniforms is essentially retractable if and only if there exist uniform retractable R-modules {U i}iÎI and R-homomorphisms M¾®+iÎI U i¾®M with ab;¹0.

A module M is called epi-retractable if every submodule of M is a homomorphic image of M. Dually, a module M is called co-epi-retractable if it contains a copy of each of its factor modules. In special case, a ring R is called co-pli (respectively, copri) if RR (respectively, RR) is co-epi-retractable. It is proved that if R is a left principal right duo ring, then every left ideal of R is an epi-retractable R-module. A co-pli strongly prime ring R is a simple ring. A left self-injective co-pli ring R is left Noetherian if and only if R is a left perfect ring. It is shown that a cogenerator ring R is a pli ring if and only if it is a co-pri ring. Moreover, if R is a left perfect ring such that every projective R-module is co-epi-retractable, then R is a quasi-Frobenius ring.

The notion of an essentially slant Toeplitz operator on the space L2 is introduced and some of the properties of the set ESTO(L2), the set of all essentially slant Toeplitz operators on L2, are investigated. In particular the conditions under which the product of two operators in ESTO(L2) is in ESTO(L2) are discussed. The notion is generalized to kth-order essentially slant Toeplitz operators.  

Let R be an arbitrary ring and N be a right R-module.  A right R-module M is called N-retractable if HomR(M,N')≠0, for any nonzero submodule N' of N. This is a generalization of the concept of retractable modules. The aim of this paper is to study of N-retractable modules, where N is an arbitrary right R-module. One of the most important results of this paper is the characterization of rings that have a module such that each module is retractable with respect to it. Also we show that the class of N-retractable modules is closed under direct sums and direct products.

There are many concepts in social sciences that are normative and controversial, and defined in different ways. Considering the principle of diverse understanding and constant dispute, it is impossible to reach to a final definition of these concepts so to be acceptable for all. At the condition of confusion and variety of interpretations in social sciences especially in political science, the contemporary British philosopher, Walter Bryce Gallie, proposed the idea of "Essentially Contested Concepts" in 1956 using the method of post-analytic philosophy. This idea seems to be a good step to reach a rational solution in understanding the important normative concepts that are constantly contested. In the first part of the paper, it is attempted to discuss Gallie’s idea and explain his criteria to accept a concept as "Essentially Contested", also different standpoints and critics over the idea are reviewed. In the second part, democracy as a key concept in political thought is selected and discussed, so to clarify whether or not democracy is an "Essentially Contested Concept". Finally, in a conclusion the importance and efficiency of Gallie’s idea in the contemporary world is emphasized and admired as a positive solution for a safe competition and a step toward pluralism and tolerance with other viewpoints.

Generalizing concepts “right Bezout ” and “principal right ideal ” of a ring R to modules, an R-module M is called n-epiretractable (resp. epi-retractable) if every n-generated submodule (resp. submodule) of MR is a homomorphic image of M. It is shown that if MR is finitely generated quasi-projective 1-epi-retractable, then EndR(M) is a right Bezout (resp. principal right ideal) ring if and only if MR is n-epi-retractable for all n³1 (resp. epiretractable). For a ring R and an infinite ordinal b³|R|, the R-module M = FÅN is epi-retractable where F is a free R-module with a basis set of cardinality b and N is a g-generated R-module with g£b. A ring R is quasi Frobenius if every injective R-module is epi-retractable. Injective modules in s[NR] are epi-retractable for every NÎs [MR] if and only if every non-zero factor ring of S is a quasi Frobenius ring where S is an endomorphism ring of a progenerator in s[MR].

Let B be the open unit ball in the n-space C-n and T-u be the Toeplitz operator on the Bergman space L-a(2)(B) with symbol u. In the present paper we:describe the essentially commuting Toeplitz operators on L-a(2)(B) with pluriharmonic symbols.

Several characterizations of a ring R is given for which any non-zero finitely generated module M is retractable in the sense that HomR(M,N) is non-zero whenever N is a non-zero submodule of M. Such rings are called finite retractable. It is shown that any ring being Morita equivalent to a commutative ring is finite retractable. Also, if the commutative ring is semi-Artinian then any non-zero module is retractable. The class of finite retractable rings is shown to be closed under Morita equivalence and finite direct products. Moreover, for a finite retractable ring R which is a right order in a ring Q, it is shown that Q is also finite retractable.