## Non-commutative noetherian rings |

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artinian ring Asano right order complete local ring correspondence defined denote direct sum direct summand division ring elements of Q endomorphism ring epimorphism essential right ideal exists factor ring finitely generated projective follows form ac groupoid hence ideal of Q idempotent implies integral R-ideal isomorphic Jacobson radical Let F Let Q M-topology maximal ideal maximal right order minimal prime Mn+1 module MS+1 nilpotent ideals noetherian non-zero ideal obtain prime ideal prime ring projective right R-ideal projective uniform right Proof prove Q has max-r R-module radical of Q regular element right Dedekind right hereditary right Ore condition right Ore domain right Q-module right quotient ring ring Q ring R/N ring with max-r simple artin ring simple ring strongly P-primary submodule subring of Q suppose symbolic powers theorem 1.1 uniform right ideal unit element units in Q units of Q zero