A commutative ring
R has Property (A) if every finitely generated ideal of
R consisting entirely of zero-divisors has a nonzero annihilator. We continue in this paper the study of rings with Property ...(A). We extend Property (A) to noncommutative rings, and study such rings. Moreover, we study several extensions of rings with Property (A) including matrix rings, polynomial rings, power series rings and classical quotient rings. Finally, we characterize when the space of minimal prime ideals of rings with Property (A) is compact.
Building on the Goodearl–Handelman–Lawrence functional representation theorem, we provide a purely topological representation (specifically, a categorical duality) for a large class of Dedekind
...σ-complete
ℓ-groups
G with order-unit
u, including all
G where
u has a finite index of nilpotence. Our duality is a far-reaching generalization of the well-known Stone duality between
σ-complete boolean algebras and basically disconnected compact Hausdorff spaces.
Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak Hochschild dimension of A is equal to the supremum of the ...Hochschild dimension of Axfor x in the decomposition space of R. Using this fact, relations are obtained among the weak Hochschild dimension of A and the weak global dimensions of A and Ae. It is shown that a central separable algebra is a biregular ring which is finitely generated over its center. A result of S. Eilenberg concerning the separability of A modulo its Jacobson radical is extended. Finally, it is shown that every homomorphic image of an algebra of weak Hochschild dimension 1 is a type of triangular matrix algebra.
PRINCIPALLY QUASI-BAER RINGS Birkenmeier, Gary F.; Kim, Jin Yong; Park, Jae Keol
Communications in algebra,
01/2001, Volume:
29, Issue:
2
Journal Article
Peer reviewed
We say a ring with unity is right principally quasi-Baer (or simply, right p.q.-Baer) if the right annihilator of a principal right ideal is generated (as a right ideal) by an idempotent. This class ...of rings includes the biregular rings and is closed under direct products and Morita invariance. The 2-by-2 formal upper triangular matrix rings of this class are characterized. Connections to related classes of rings (e.g., right PP, Baer, quasi-Baer, right FPF, right GFC, etc.) are investigated. Examples to illustrate and delimit the theory are provided.
On N–Flat Rings Raida Mahmood; Husam Mohammad
AL-Rafidain journal of computer sciences and mathematics,
07/2011, Volume:
8, Issue:
1
Journal Article
Peer reviewed
Open access
Let I be a right ideal of R, then R / I is a right N–flat if and only if for each a Î I, there exists b Î I and a positive integer n such that an ≠ 0 and an = ban. In this paper, we first give and ...develop various properties of right N-flat rings, by which, many of the known results are extended. Also, we study the relations between such rings and regular, p-biregular ring.
On sπ-Weakly Regular Rings Raida Mahmood; Abdullah Abdul-Jabbar
AL-Rafidain journal of computer sciences and mathematics,
12/2008, Volume:
5, Issue:
2
Journal Article
Peer reviewed
Open access
A ring R is said to be right(left) sp-weakly regular if for each a Î R and a positive integer n, aÎ aR aR (aÎ R aR a). In this paper, we continue to study sp-weakly regular rings due to R. D. ...Mahmood and A. M. Abdul-Jabbar 8. We first consider properties and basic extensions of sp-weakly regular rings, and we give the connection of sp-weakly regular, semi p-regular and p-biregular rings.
Baer–Stone Shells Knoebel, Arthur
Sheaves of Algebras over Boolean Spaces,
10/2011
Book Chapter
Here are some applications of the theory of the previous chapter. In the first section, it is proven that, for every Baer–Stone half-shell that is two-sided and unital, there is a reduced and ...factor-transparent sheaf over a Boolean space that represents the half-shell and and has stalks with no divisors of zero. With all that has been done in previous chapters, the proof is relatively short. Just as the results of the previous chapters may be cast into categories, so we restate this result as the equivalence of two categories. In the second section are two more applications. Each von Neumann regular, commutative and unital ring is isomorphic to the ring of all global sections of a sheaf of fields over a Boolean space. And every biregular ring is isomorphic to the ring of all global sections of a sheaf with simple stalks over a Boolean space. These results extend to half-shells.
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