Communicated by B. MondBaer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then # [ X ] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. J0ndrup [5] who proved stability for commutative P.P.-rings via localizations -a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.We will assume throughout that rings have a unit. For convenience we call a ring reduced if it has no nonzero nilpotent elements. In a reduced ring R left and right annihilators coincide for any subset U of R, hence we let ann R The key lemma is the following characterization of zero divisors in R[X~\ when R is a reduced ring.
ABSTRACT. This paper studies maximal quotient rings of semiprime P. I.-rings; such rings are regular, self-injective and satisfy a polynomial identity.We show that the center of a regular self-injective ring is regular self-injective;this enables us to establish that the center of the maximal quotient ring of a semiprime P. I.-ring R is the maximal quotient ring of the center of R, as well as some other relationships. We give two decompositions of a regular self-injective ring with a polynomial identity which enable us to show that such rings are biregular and are finitely generated projective modules over their center.
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