We will consider MULTIPLICATION OPERATORS on Hilbert spaces of analyticfunctions on a domain $\omega \subset \cmark $. We determine the commutants of certainMULTIPLICATION OPERATORS with adjoints in a Cowen-Douglas class operator.

In this paper, we determine the structure of the space of multipliers of the range of a composition operator Cj that induces by the conditional expectation between two Lp(å) spaces.

In this paper the conditional multipliers acting between Lp spaces are characterized by using some properties of conditional expectation operator. Also, we determine the essential norm of uCj on Lp for 1 < p < ¥.

In this paper we introduce the concept of MULTIPLICATION-like modules and we obtain some related results. We show that an R-module M is MULTIPLICATION-like if and only if for each ideal I of R, I = (IM: R M). We prove that any MULTIPLICATION-like module is faithful and r-MULTIPLICATION. So we get that any at and MULTIPLICATION-like module is faithfully at.

Let M be a lattice module over the multiplicative lattice L. An L-module M is called a MULTIPLICATION lattice module if for every element NÎM there exists an element aÎL such that N=a1M. Our objective is to investigate properties of prime elements of MULTIPLICATION lattice modules.

In this paper we introduce the notion of MULTIPLICATION ideals in Γ-rings and we obtain some characterizations for MULTIPLICATION ideals in Γ-rings.

We state several conditions under which comultiplica-tion and weak coMULTIPLICATION modules are cyclic and study strong coMULTIPLICATION modules and coMULTIPLICATION rings. In particu-lar, we will show that every faithful weak coMULTIPLICATION module having a maximal submodule over a reduced ring with a  nite in-decomposable decomposition is cyclic. Also we show that if M is an strong coMULTIPLICATION R-module, then R is semilocal and M is  nitely cogenerated. Furthermore, we de ne an R-module M to be p-coMULTIPLICATION, if every nontrivial submodule of M is the annihilator of some prime ideal of R containing the annihila-tor of M and give a characterization of all cyclic p-coMULTIPLICATION modules. Moreover, we prove that every p-coMULTIPLICATION module which is not cyclic, has no maximal submodule and its annihilator is not prime. Also we give an example of a module over a Dedekind domain which is not weak coMULTIPLICATION, but all of whose local-izations at prime ideals are coMULTIPLICATION and hence serves as a counterexample to [11, Proposition 2. 3] and [12, Proposition 2. 4].

Let R be a commutative ring with identity and M be a unitary R-module. An R-module M is called a MULTIPLICATION module if for every submodule N of M there exists an ideal I of R such that N=IM. It is shown that over a Noetherian domain R with dim(R)≤ 1, MULTIPLICATION modules are precisely cyclic or isomorphic to an invertible ideal of R. Moreover, we give a characterization of finitely generated MULTIPLICATION modules.