On quasi nil-injective modules

Abstract. Module \(M\) is called quasi nil-injective if for each \(m\in Nil(M)\) and each homomorphism \(f: mR\rightarrow M\), there exists a homomorphism \(\bar{f}: M\rightarrow M\) such that \(\bar{f}(x)=f(x)\) for every \(x\in mR\). In this paper, we first obtain some characterizations of the class of quasi nil-injective modules and some known results can be deduced from these characteristics. Next, we apply to ring and obtain some properties of a quasi nil-injective rings. We proved that a ring \(R\) is semiprime if only if every right \(R\)-module (cyclic) is quasi nil-\(R\)-injective.