It is known that each word of length n contains at most n+1 distinct
palindromes. A finite rich word is a word with maximal number of palindromic
factors. The definition of palindromic richness can be naturally extended to
infinite words. Sturmian words and Rote complementary symmetric sequences form
two classes of binary rich words, while episturmian words and words coding
symmetric d-interval exchange transformations give us other examples on
larger alphabets. In this paper we look for morphisms of the free monoid, which
allow to construct new rich words from already known rich words. We focus on
morphisms in Class Pret. This class contains morphisms injective on the
alphabet and satisfying a particular palindromicity property: for every
morphism φ in the class there exists a palindrome w such that
φ(a)w is a first complete return word to w for each letter a. We
characterize Pret morphisms which preserve richness over a binary
alphabet. We also study marked Pret morphisms acting on alphabets with
more letters. In particular we show that every Arnoux-Rauzy morphism is
conjugated to a morphism in Class Pret and that it preserves richness