M. Roueentan - Lamerd Higher Education Center, Lamerd, Iran.
M. Ershad - Department of Mathematics, College of Sciences,
M. A. Naghipoor - Department of Mathematics, Jahrom University,

This paper is a continuation of recent researches concerning generalization of injectivity of acts over moniods, namely, C-injectivity and InD-injectivity. We introduce new kinds of injectivity, such as, LC-injectivity and CQ-injectivity. Classications of monoids by properties of these kinds of injective acts are presented. It is proved that a monoid S is completely (cyclic) injective if and only if it is completely quasi (CQ-) injective. Some results on quasi-injective acts are proved. Also new characterizations for right hereditary monoids and right PP-monoids are given.

Locally cyclic (weakly) injective act, Indecomposable (weakly) injective act, C- injective act