In ring theory one can give several approaches to the introduction of the concept of semisimplicity and a large number of equivalent formulations of this concept [5, 13]. Analogues of some of these formulations have been made, and studied, for semigroups [4, 6, 7, 8, 10, 11, 12] in terms of ideals or congruence relations. It seems possible that suitable and effective analogues can be made for each of these ring theoretical formulations in terms of congruence relations. However, unlike the situation for rings most of these analogues give inequivalent formulations in semigroups. One of the weakest of these is the nonexistence of a proper, essential right congruence. A right congruence ~ in a semigroup S is essential if for any right congruence a we have Q N a = z (the identity relation) implies a = z. The main result of this paper is a characterization of semigroups with the d.c.c, on right ideals and having no proper essential right ideals and having no proper essential right congruences. The first step in this characterization is a description of the lattice of right ideals in such a semigroup. Our results of this description in Section 2 should be compared with the work of Feller and Gantos [2] and Fountain [3]. While the class of semigroups studied in these two papers are defined quite differently than the class in this paper there is a striking similarity in the results; thus, suggesting a common area of investigation. In the subsequent sections the main technique used is the examination of a selection of proper right congruences and the implications on the multiplicative properties obtained from the assumption of nonessentiality.