In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let R be a ring and R[x] be the polynomial ring over R with x the indeterminate. We show that eRe is right quasi-duo for if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[] is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if R[x] is right quasi-duo then the Jacobson radical of R[x] is N(R)[x] and so the conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where N(R) is the set of all nilpotent elements in R. Next we prove that if the polynomial rins R[x], over a reduced ring R with X 2, is right quasi-duo, then R is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.

quasi-duo ring;polynomial ring;Jacobson radical;commutative ring;

English