Reduction of Finite Baer ∗-Rings

1972 ◽  
pp. 186-209
Author(s):  
Sterling K. Berberian
Keyword(s):  
Baer Rings

Partial orders on the power sets of Baer rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050011 ◽  
Author(s):  
B. Ungor ◽  
S. Halicioglu ◽  
A. Harmanci ◽  
J. Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Baer Ring ◽  
Power Set ◽  
Von Neumann Regular ◽  
Baer Rings ◽  
The Right

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

scholarly journals A note on extensions of Baer and P. P. -rings

1974 ◽  
Vol 18 (4) ◽  
pp. 470-473 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Left Ideal ◽  
Polynomial Identity ◽  
Polynomial Rings ◽  
Nilpotent Elements ◽  
Baer Ring ◽  
Baer Rings ◽  
Left Annihilator

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

Baer ∗-Rings

1972 ◽  
Author(s):  
Sterling K. Berberian
Keyword(s):  
Baer Rings

αcc-Baer Rings

2015 ◽  
Vol 65 (2) ◽  
Author(s):  
R. E. Carrera ◽  
Iberkleid ◽  
Lafuente-Rodriguez ◽  
W. W. McGovern
Keyword(s):  
Semiprime Ring ◽  
Infinite Cardinal ◽  
Baer Ring ◽  
Ring Of Quotients ◽  
Baer Rings

AbstractLet α denote an infinite cardinal or ∞ which is used to signify no cardinal constraint. This work introduces the concept of an αcc-Baer ring and demonstrates that a commutative semiprime ring A with identity is αcc-Baer if and only if Spec(A) is αcc-disconnected. Moreover, we prove that for each commutative semprime ring A with identity there exists a minimum αcc-Baer ring of quotients, which we call the αcc-Baer hull of A. In addition, we investigate a variety of classical α-Baer ring results within the contexts of αcc-Baer rings and apply our results to produce alternative proofs of some classical results such as A is α-Baer if and only if Spec(A) is α-disconnected. Lastly, we apply our results within the contexts of archimedean f-rings.

Feebly Baer Rings and Modules

2014 ◽  
Vol 42 (10) ◽  
pp. 4281-4295
Author(s):  
M. Tamer Koşan ◽  
Tsiu-Kwen Lee ◽  
Yiqiang Zhou
Keyword(s):  
Baer Rings

A NOTE ON PRINCIPALLY QUASI-BAER RINGS

2002 ◽  
Vol 30 (8) ◽  
pp. 3885-3890 ◽  
Author(s):  
Zhongkui Liu
Keyword(s):  
Baer Rings

scholarly journals A note on commutative Baer rings

1972 ◽  
Vol 14 (3) ◽  
pp. 257-263 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Semiprime Ring ◽  
Algebraic Approach ◽  
Direct Decomposition ◽  
Prime Ideals ◽  
Baer Ring ◽  
Baer Rings ◽  
New Construction ◽  
Minimal Prime

In this note we study commutative Baer rings, uniting the abstract algebraic approach with the approach of [3] using minimal prime ideals. Some new characterisations of this class of rings are obtained, relations between the minimal prime ideals of a commutative Baer ring B and its algebra EB of idempotents are considered, and some results concerning the direct decomposition of commutative Baer rings are given. We then study Baer ideals, and finally state without proof a new construction of the Baer extension of a commutative semiprime ring.

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