On Locally Injective Acts over Monoids

In this paper, a new generalization of injective acts over a monoid, namely locally injective, are introduced and studied. Using examples, we exhibit that this generalization is proper. Various characterizations of locally injective acts are provided. For example, we show that an act is locally injective if it is locally retracted from each of its extensions. Additionally, we generalize some of the well-known results of injective acts. We prove that every locally injective act has a zero element and is divisible. The relationship among locally injective acts and other known generalizations if injective acts like as principally S-acts, finitely injective S-acts and divisible S-acts are discussed. In fact, we get the following implications: injective S-acts finitely injective S-acts locally injective S-acts principally injective S-acts. Moreover, we assert that locally injective property is closed under product and retracts. Finally, we characterize a monoid by their local injectivity.