Some results on rings with chain conditions (Q1071095)

If A is a right noetherian ring such that A/P is right artinian for any prime ideal P, then A is right artinian if and only if its additive group has no direct summand isomorphic to the infinite cyclic group. A right noetherian ring is right artinian if and only if for any prime ideal P, A/P is a simple ring having a non-zero socle and for any homomorphic image A' of A the minimum condition holds for such principal right ideals of A' which are contained in the annihilator of A'. Also strictly linearly compact rings which are topologically noetherian, are characterised, and as a corollary the author gets that for any strictly linearly compact ring A the maximal divisible torsion ideal of A is contained in the left annihilator of A. If A is a strictly linearly compact ring such that the intersection of the closures of all finite powers of the Jacobson radical J is 0 and \(A/cl(J^ 2)\) is commutative, then A is algebraically and topologically isomorphic to a complete direct sum of strictly linearly compact local rings with identity and of a strictly linearly compact Jacobson radical ring.