Rigid left Noetherian rings

Let R be an associative ring. A map σ : R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R . Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid ring R with the additive group R+ of finite Prüfer rank. A characterization for rigid rings of finite rank was obtained by the author in [1]. Some aspects of a ring rigidity has been studied by Suppa [12, 13], Friger [5], and the author [2]. In this paper, we study rigid left Noetherian rings and prove the following theorem.