Hong Kee Kim

p.p. rings and generalized p.p. rings

Abstract This paper concerns two conditions, called right p.p. and generalized right p.p., which are generalizations of Baer rings and von Neumann regular rings. We study the subrings and extensions of them, adding proper examples and counterexamples to some situations and questions that occur naturally in the process of this paper.

ON A GENERALIZATION OF THE MCCOY CONDITION

We in this note consider a new concept, so called …-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of …-McCoy rings contain upper (lower) trian- gular matrix rings and many kinds of full matrix rings. We first study the basic structure of …-McCoy rings, observing the relations among …-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and (…-)regular rings. It is proved that the n by n full matrix rings (n ‚ 2) over reduced rings are not …-McCoy, finding …-McCoy matrix rings over non-reduced rings. It is shown that the …-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of …-McCoy rings are also examined. 1. Basic properties of …-McCoy rings

NILRADICALS OF POWER SERIES RINGS AND NIL POWER SERIES RINGS

Klein proved that polynomial rings over nil rings of bounded index are also nil of bounded index; while Puczylowski and Smoktunowicz described the nilradical of a power series ring with an indeterminate. We extend these results to those with any set of commuting indeterminates. We also study prime radicals of power series rings over some class of rings containing the case of bounded index, flnding some examples which elaborate our arguments; and we prove that R is a PI ring of bounded index then the power series ring R((X)), with X any set of indeterminates over R, is also a PI ring of bounded index, obtaining the Kleins result for polynomial rings as a corollary.

Reflexive property on nil ideals

We in this note consider the reflexive ring property on nil ideals, introducing the concept of a nil-reflexive ring as a generalization of the reflexive ring property. We will call a ring R nil-reflexive if IJ = 0 implies JI = 0 for nil ideals I,J of R. The polynomial and the power series rings over a right Noetherian ring (or an NI ring) R are shown to be nil-reflexive if (aRb)2 = 0 implies aRb = 0 for all a,b ∈ N(R). We further investigate the structure of nil-reflexive rings, related to various sorts of ring extensions which have roles in ring theory.

On a Hereditary Radical Property Relating to the Reducedness

In this note, a hereditary radical property, called homomorphically reduced rings, is introduced, observed, and applied. The dual concept of this property is also studied with the help of Courter, proving that any ring R (possibly without identity) has an ideal S such that S/K is not homomorphically reduced for each proper ideal K of S; and if L is an ideal of R with L ⊊ S, then L/H is homomorphically reduced for some ideal H of R with H ⊊ L. The concept of the homomorphical reducedness is shown to be equivalent to the left (right) weak regularity and the (strong) regularity for one-sided duo rings. It is proved that homomorphically reduced rings have several useful properties similar to those of (weakly) regular rings. It is proved that the homomorphical reducedness can go up to classical quotient rings. It is shown that if R is a reduced right Ore ring with the ascending chain condition (ACC) for annihilator ideals, then the maximal right quotient ring of R is strongly regular (hence homomorphically reduced).

ON CONDITIONS PROVIDED BY NILRADICALS

A ring R is called IFP, due to Bell, if ab = 0 implies aRb = 0 for a,b 2 R. Huh et al. showed that the IFP condition is not preserved by polynomial ring extensions. In this note we concentrate on a generalized condition of the IFPness that can be lifted up to polynomial rings, intro- ducing the concept of quasi-IFP rings. The structure of quasi-IFP rings will be studied, characterizing quasi-IFP rings via minimal strongly prime ideals. The connections between quasi-IFP rings and related concepts are also observed in various situations, constructing necessary examples in the process. The structure of minimal noncommutative (quasi-)IFP rings is also observed.

WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL

Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not di-cult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we flrst pro- vide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring R is weakly duo under each of the fol- lowing conditions: (1) R is semilocal with nil Jacobson radical; (2) R is locally flnite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring R, obtaining that i(R) is flnite and a i(R) R = Ra i(R) = Ra i(R) R for all a 2 R, where i(R) is the index (of nilpotency) of R. Note that one-sided Artinian rings and locally flnite rings are strongly …-regular. Thus we also observe connections between strongly …- regular weakly right duo rings and related rings, constructing avail- able examples.

On a Generalized Finite Intersection Property

We continue the study of the right finite intersection property under a weaker condition on annihilators, introducing the concept of generalized right finite intersection property (simply, generalized right FIP). We observe the structure of rings with the generalized right FIP and examine the generalized right FIP for various kinds of basic extensions of rings with the property. We show that the generalized right FIP does not go up to polynomial rings, and that the 2-by-2 full matrix ring over a domain has the generalized right FIP. In the process, we also obtain an equivalent condition for which a nonzero polynomial, over the ring of integers modulo n ≥ 2, is a non-zero-divisor.

A Note on GP-Injectivity

New characteristic properties of left GPGV-rings are given. It is shown that if R is a left GPGV-ring, then for any nonzero element a in R, there is a positive integer n such that an≠ 0 and (RaR+ l(an))⊕ L=R for some left ideal L contained in Soc(RR). As a corollary of this result, we are able to give a positive answer to a question raised by Yue Chi Ming.

On a ring structure related to annihilators

In this note, we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (respectively, right) ZAFS property is shown to pass also to classical left (respectively, right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.