Chan Yong Hong

Ore extensions of Baer and p.p.-rings

Abstract We investigate Ore extensions of Baer rings and p.p.-rings. Let α be an endomorphism and δ an α-derivation of a ring R. Assume that R is an α-rigid ring. Then (1) R is a Baer ring if and only if the Ore extension R[x;α,δ] is a Baer ring if and only if the skew power series ring R[[x;α]] is a Baer ring, (2) R is a p.p.-ring if and only if the Ore extension R[x;α,δ] is a p.p.-ring.

On Skew Armendariz Rings

Abstract For a ring endomorphism α, we introduce α-skew Armendariz rings which are a generalization of α-rigid rings and Armendariz rings, and investigate their properties. Moreover, we study on the relationship between the Baerness and p.p.-property of a ring R and these of the skew polynomial ring R[x; α] in case R is α-skew Armendariz.

On minimal strongly prime ideals

In this paper we give some characterizations of a ring Rwhose unique maximal nil ideal N r (R) coincides with the set of all its nilpotent elements N(R) by using its minimal strongly prime ideals.

On weak π-regularity of rings whose prime ideals are maximal

We investigate, in this paper, the connections between the weak π-regularity and the maximality of prime ideals in 2-primal rings, right quasi-duo rings and PI-rings, respectively.

NILPOTENT IDEALS IN POLYNOMIAL AND POWER SERIES RINGS

Given a ring R and polynomials f(x), g(x) ∈ R[x] satisfying f(x)Rg(x) = 0, we prove that the ideal generated by products of the coefficients of f(x) and g(x) is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if I ≤ R[x] is a left T-nilpotent ideal, then the ideal formed by the coefficients of polynomials in I is also left T-nilpotent.

Exchange rings and their extensions

Abstract A ring R is called to be exchange if the right regular module R R has finite exchange property. We continue in this paper the study of exchange rings by several authors. In particular, we investigate the von Neumann regularity of exchange rings. In addition, we also study whether the exchange property is inherited by some extensions of exchange rings.

The McCoy Condition on Noncommutative Rings

McCoy proved in 1957 [12] that if a polynomial annihilates an ideal of polynomials over any ring then the ideal has a nonzero annihilator in the base ring. We first elaborate this McCoys famous theorem further, expanding the inductive construction in the proof given by McCoy. From the proof we can naturally find nonzero c, with f(x)c = 0, in the ideal of R generated by the coefficients of g(x), when f(x), g(x) are nonzero polynomials over a commutative ring R with f(x)g(x) = 0; from which we also obtain a kind of criterion for given a polynomial to be a zero divisor. Based on these results we extend the McCoys theorem to noncommutative rings, introducing the concept of strong right McCoyness. The strong McCoyness is shown to have a place between the reversibleness (right duoness) and the McCoyness. We introduce a simple way to construct a right McCoy ring but not strongly right McCoy, from given any (strongly) right McCoy ring. If given a ring is reversible or right duo, then the polynomial ring over it is proved to be strongly right McCoy. It is shown that the (strong) right McCoyness can go up to classical right quotient rings.

Counterexamples on baer rings

In this paper we give counterexamples to the following questions: (1) Are commutative reduced rings Baer?, (2) Are commutative von Neumann regular rings Baer?, (3) Are reduced rings with center Baer also Baer?, and (4) Are prime Pi-rings Baer? Moreover we consider some conditions under which the answers of them are affirmative.

ORE EXTENSIONS OF QUASI-BAER RINGS

We first study the quasi-Baerness of R[x; σ, δ] over a quasi-Baer ring R when σ is an automorphism of R, obtaining an affirmative result. We next show that if R is a right principally quasi-Baer ring and σ is an automorphism of R with σ(e) = e for any left semicentral idempotent e ∈ R, then R[x; σ, δ] is right principally quasi-Baer. As a corollary, we have that R[x; δ] over a right principally quasi-Baer ring R is right principally quasi-Baer. Finally, we give conditions under which the quasi-Baernesses (right principal quasi-Baernesses) of R and R[x; σ, δ] are equivalent.

Extensions of Generalized Reduced Rings

Anderson and Camillo studied the class of rings satisfying ZCn for n ≥ 2, which is a generalization of reduced rings. In this paper, we continue the study of such rings. We observe several extensions of rings satisfying ZCn. Rings satisfying the zero insertion property for n (simply, ZIn), which is a generalization of ZCn, are also introduced. In particular, we prove that every ring satisfying ZIn for some n ≥ 2 is a 2-primal ring. Furthermore, if R is an Armendariz ring satisfying ZIn for n ≥ 2, then the polynomial ring R[x] over R also satisfies ZIn.