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We study the structure of rings which satisfy the von Neumann regularity of commutators, and call a ring R C-regular if a b − b a ∈ (a b − b a) R (a b − b a) for all a , b in R. For a C-regular ring R , we prove J (R [ X ]) = N ∗ (R [ X ]) = N ∗ (R) [ X ] = W (R) [ X ] ⊆ Z (R [ X ]) , where J (A) , N ∗ (A) , W (A) , Z (A) are the Jacobson radical, upper nilradical, Wedderburn radical, and center of a given ring A , respectively, and A [ X ] denotes the polynomial ring with a set X of commuting indeterminates over A ; we also prove that R is semiprime if and only if the right (left) singular ideal of R is zero. We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular, from any given ring. Moreover, for a C-regular ring R , the following are proved to be equivalent: (i) R is Abelian; (ii) every prime factor ring of R is a duo domain; (iii) R is quasi-duo; and (iv) R / W (R) is reduced. [ABSTRACT FROM AUTHOR]

Jacobson investigated the structure of rings with the property that some power of each element is central, in the procedure of the study of commutativity. In this paper, we consider a class of rings in which this property occurs only for central elements, and such rings are called ppnc. We first prove that a noncommutative ppnc ring is infinite, and that a ppnc ring is commutative when it is a K-ring or a locally finite ring. We next study the structure of ppnc rings, and the relation between ppnc rings and related concepts (for example, K-rings, commutative rings and NI rings), through matrix rings, polynomial rings, right quotient rings and direct products. [ABSTRACT FROM AUTHOR]

We continue the study of the reflexivity of ideals, introduced by Mason, and extend this notion to the subsets in rings. We first construct the smallest reflexive ideal containing S from any proper ideal S of any given ring R ; by which we can construct reflexive ideals but not semiprime in a kind of noncommutative ring. A subset T of a ring R is called reflexively closed if a R b ⊆ T for a , b ∈ R implies b R a ⊆ T , checking that a ring R is symmetric if and only if the right (left) annihilator of a is reflexively closed for any a ∈ R. We prove that the set of all nilpotent elements in a ring R is reflexively closed if and only if a R is nil for any nilpotent element a in R ; and that the Köthe's conjecture holds if and only if the union (sum) of the upper nilradical and any nil right ideal is reflexively closed. We provide another process to show that the set of all nilpotent elements of the polynomial ring over an NI ring need not be reflexively closed. [ABSTRACT FROM AUTHOR]

The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements. Such rings shall be called right exp-DR. We investigate the structures of group rings, right quotient rings, matrix rings and (skew) polynomial rings, through the study of right exp-DR rings. In addition, we provide a method of constructing finite non-abelian p -groups for any prime p. [ABSTRACT FROM AUTHOR]

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. We in this note study rings with the reflexivity whose axis is given by maximal ideals (simply, an RM ring) which are a generalization of symmetric rings. It is first shown that the reflexivity of a ring and the RM ring property are independent of each other, noting that both of them are generalizations of ideal-symmetric rings. We connect RM rings with reflexive rings in various situations raised naturally in the procedure. As a generalization of RM rings, we also study the structure of the reflexivity with the maximal ideal axis on idempotents (simply, an RMI ring) and then investigate the structure of minimal non-Abelian RMI rings (with or without identity) up to isomorphism. [ABSTRACT FROM AUTHOR]