Integrally closed Noetherian rings

All rings are assumed to be associative and unital. A module is said to be integrally dosed (semiinjective) if all the endomorphisms of its finitely-generated submodules (its submodules) extend to the whole modules. A ring is said to be right-invariant (left-invariant) if all its right (left) ideals are ideals. A ring without non-zero nilpotent elements is said to be reduced. A ring is said to arithmetic (leftdistributive) if Α η (B + Q = Α η Β +Α η C for any ideals (left ideals) A, B, and C in it. The expressions Noetherian ring, invariant ring, and so on, mean that the corresponding left and right conditions hold.