ON GENERALIZATION OF EXTENDING ACTS AND M-JECTIVE ACTS

An S-act M_s is called a generalized extending act (for short a GE-act) if the following condition is satisfied: If M_s = M_1 ⋃ ̇ M_2, and X is subact of M_s , then there exist C_i is a retract of M_i (i = 1, 2) such that C_1 ⋃ ̇ C_2 is a complement of X inM_s. In this article, the notion of generalized extending S-act is introduced and studied as a concept of generalizing extending act which was presented by the author. Some properties of such acts in analogy with the known properties for extending acts are illustrated .Besides, the author has introduced in a diagram of acts and homomorphisms, the concept of generalized of quasi injective which is also representing a generalization of M-injective acts. Here we introduce the concept of M-jective acts, which is a generalization of the concept of M-injectivity. An S-act Y is called X-jective if every complement Z of Y in M_s is a retract, where M_s = X⋃ ̇Y. The concept of M-jective acts is used here to solve the problem of finding a necessary and sufficient condition for a direct sum of extending acts to be extending. Indeed, we show that relative jectivity is necessary and sufficient for a direct sum of two extending acts to be extending as in module theory. Some properties and characterizations of generalizing extending act and M-jective act are illustrated. Conditions on which subact inherit the property of generalizing extending act were demonstrated. The relationship among extending act and generalizing extending act, act with condition 〖(C〗_1^*) and generalizing extending act was elucidated. Conclusions and discussion of this work were clarified in the last section.