P.P. Rings and Generalized P.P. Rings (original) (raw)

International Journal of Pure and Apllied Mathematics, 2015

A ring R is called generalized right Baer if for any non-empty subset S of R, the right annihilator r R (S n) is generated by an idempotent for some positive integer n. Generalized Baer rings are special cases of generalized PP rings and a generalization of Baer rings. In this paper, many properties of these rings are studied and some characterizations of von Neumann regular rings and PP rings are extended. The behavior of the generalized right Baer condition is investigated with respect to various constructions and extensions and it is used to generalize many results on Baer rings and generalized right PP-rings. Some families of generalized right Baer-rings are presented and connections to related classes of rings are investigated.

A Generalization of Von Neumann Regular Rings

AL-Rafidain Journal of Computer Sciences and Mathematics, 2009

In this paper, we introduce a new ring which is a generalization of Von Neumann regular rings and we call it a centrally regular ring. Several properties of this ring are proved and we have extended many properties of regular rings to centrally regular rings. Also we have determined some conditions under which regular and centrally regular rings are equivalent.

Generalized π-Baer rings

TURKISH JOURNAL OF MATHEMATICS

We call a ring R generalized right π -Baer, if for any projection invariant left ideal Y of R , the right annihilator of Y n is generated, as a right ideal, by an idempotent, for some positive integer n , depending on Y . In this paper, we investigate connections between the generalized π -Baer rings and related classes of rings (e.g., π -Baer, generalized Baer, generalized quasi-Baer, etc.) In fact, generalized right π -Baer rings are special cases of generalized right quasi-Baer rings and also are a generalization of π -Baer and generalized right Baer rings. The behavior of the generalized right π -Baer condition is investigated with respect to various constructions and extensions. For example, the trivial extension of a generalized right π -Baer ring and the full matrix ring over a generalized right π -Baer ring are characterized. Also, we show that this notion is well-behaved with respect to certain triangular matrix extensions. In contrast to generalized right Baer rings, it is shown that the generalized right π -Baer condition is preserved by various polynomial extensions without any additional requirements. Examples are provided to illustrate and delimit our results.

A certain N -Generalized Principally Quasi-Baer Subring of the

For a fixed positive integern , we say a ring with identity is n-generalized right principally quasi-Baer, if for any principal right ideal I ofR , the right annihilator of n I is generated by an idempotent. This class of rings includes the right principally quasi-Baer rings and hence all prime rings. A certain n-generalized principally quasi-Baer subring of the matrix ring () n MR are studied, and connections to related classes of rings (e.g., p.q.-Baer rings and n-generalized p.p. rings) are considered 1 . quasi-Baer ring which is not a Baer ring by [27] and [21, p.17]. The nn ´ ( 1 n > ) upper triangular matrix ring over a domain which is not a division ring is quasi-Baer but not 1. 2000 Mathematical Subject Classification. 16D15; 16D40; 16D70.

On Rings Whose Right Annihilators Are Bounded

Glasgow Mathematical Journal, 2009

Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterex...

A Characterization of Baer-Ideals

2014

An ideal I of a ring R is called right Baer-ideal if there exists an idempotent e 2 R such that r(I) = eR. We know that R is quasi-Baer if every ideal of R is a right Baer-ideal, R is n-generalized right quasi-Baer if for each I E R the ideal In is right Baer-ideal, and R is right principaly quasi-Baer if every principal right ideal of R is a right Baer-ideal. Therefore the concept of Baer ideal is important. In this paper we investigate some properties of Baer-ideals and give a characterization of Baer-ideals in 2-by-2 generalized triangular matrix rings, full and upper triangular matrix rings, semiprime ring and ring of continuous functions. Finally, we find equivalent conditions for which the 2-by-2 generalized triangular matrix ring is right SA.

On the structure of certain von Neumann regular and π-regular rings with prime center

International Journal of Algebra

We study the structure of certain von Neumann regular rings and π-regular rings with certain constraints such as having a prime and other constraints. For example, we prove that a π-regular ring with prime center is strongly π-regular, and other related results are also proved. An example is also given to illustrate our result.

A Note on Extensions of Principally Quasi-Baer Rings

Taiwanese Journal of Mathematics, 2008

Let R be a ring with unity. It is shown that the formal power series ring R[[x]] is right p.q.-Baer if and only if R is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join.

Von Neumann Regular and Related Elements in Commutative Rings

Algebra Colloquium, 2012

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.