Tai Keun Kwak

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.

The McCoy Condition on Skew Polynomial Rings

Based on a theorem of McCoy on commutative rings, Nielsen called a ring R right McCoy if, for any nonzero polynomials f(x), g(x) over R, f(x)g(x) = 0 implies f(x)r = 0 for some 0 ≠ r ∈ R. In this note, we consider a skew version of these rings, called σ-skew McCoy rings, with respect to a ring endomorphism σ. When σ is the identity endomorphism, this coincides with the notion of a right McCoy ring. Basic properties of σ-skew McCoy rings are observed, and some of the known results on right McCoy rings are obtained as corollaries.

On Extended Reversible Rings

An endomorphism α of a ring R is called right reversible if whenever ab = 0 for a, b ∈ R, then bα(a) = 0. A ring R is called right α-reversible if there exists a right reversible endomorphism α of R. The notion of an α-reversible ring is a generalization of α-rigid rings as well as an extension of reversible rings. We study characterizations of α-reversible rings and their related properties including extensions. The relationship between α-reversible rings and generalized Armendariz rings is also investigated. Several known results relating to α-rigid and reduced rings can be obtained as corollaries of our results.

Reflexive Property of Rings

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.

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

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.

Weak Quasi-Armendariz Rings

In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/〈xn〉 is weak quasi-Armendariz, where R[x] is the polynomial ring over R and 〈xn〉 is the ideal of R[x] generated by xn. Various properties of weak quasi-Armendariz rings are also observed.