The paper describes some basic properties of a class of module maps of Hilbert C*-modules.
In Section 1 ideal submodules are considered and the canonical Hilbert C*-module structure on the quotient of a Hilbert C*-module over an ideal submodule is described.
Given a Hilbert C*-module V, an ideal submodule Vinss, and the quotient V/Vinss, canonical morphisms of the corresponding C*-algebras of adjointable operators are discussed.
In the second part of the paper a class of module maps of Hilbert C*-modules is introduced.Given Hilbert C*-modules V and W and a morphism varphi:assrightarrowbss of the underlying cez-algebras, a map Phi:VrightarrowW belongs to the class under consideration if it preserves inner products modulo varphi: langlePhi(x),Phi(y)rangle=varphi(langlex,yrangle) for all x,yinV.
It is shown that each morphism Φ of this kind is necessarily a contraction such that the kernel of Φ is an ideal submodule of V. A related class of morphisms of the corresponding linking algebras is also discussed