A remark on the intersection of powers of the Jacobson radical

If A is a left Noetherian, right distributive ring, then \( \bigcap\limits_{k = 1}^\infty {{{\left( {J(A)} \right)}^k} = 0} \).     

Modules over formal matrix rings

This work contains some new and known results on modules over formal matrix rings. The main results are presented with proofs.     

Rings with projective principal right ideals

It has been proved that if A is a right-distributive ring, algebraic over its center, and whose principal ideals are projective, then A is a left-distributive ring.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 861–863, June, 1990.     

Modules over serial rings

This paper continues the study of Noetherian serial rings. General theorems describing the structure of such rings are proved. In particular, some results concerning π-projective and π-injective modules over serial rings are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 880–892 June, 1999.     

Distributive semiprime rings

It is proved that a right distributive semiprime PI ringA is a left distributive ring and for each elementx ∈A there is a positive integern such thatx ⁿ A=Ax ⁿ . We describe both right distributive right Noetherian rings algebraic over the center of the ring and right distributive left Noetherian PI rings. We also characterize rings all of whose Pierce stalks are right chain right Artin rings. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 736–761, November, 1995.     

On quaternion algebras

It is proved that the distributiveness of the right ideals lattice for a quaternion algebra over a commutative ring A is equivalent to the following property: the equation x²+y²+z²=0 is uniquely solvable in the field A/M for any maximal ideals M of A, the lattice of the ideals of A being distributive. Bibliography: 5 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 209–214, 1994.