Let R be a ring with a derivation 5 and R（（x-1;5）） denote the pseudo- differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R（（x-1; 5））. ...Let R be a ring with a derivation 5 and R（（x-1;5）） denote the pseudo- differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R（（x-1; 5））. Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R（（x-1; 5）） is Baer （resp., quasi-Baer, PP, right zip） if and only if R is a Baer （resp., quasi-Baer, PP, right zip） ring. For a 5-weakly rigid ring R, R（（x-1;5）） is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.展开更多
文摘Let R be a ring with a derivation 5 and R（（x-1;5）） denote the pseudo- differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R（（x-1; 5））. Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R（（x-1; 5）） is Baer （resp., quasi-Baer, PP, right zip） if and only if R is a Baer （resp., quasi-Baer, PP, right zip） ring. For a 5-weakly rigid ring R, R（（x-1;5）） is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.
