scholarly journals A characterization of the maximal Von Neumann regular ideal in Jordan rings

1969 ◽  
Vol 12 (2) ◽  
pp. 227-230
Author(s):  
Chester E. Tsai
Keyword(s):  
Von Neumann ◽  
Regular Ideal ◽  
Von Neumann Regular

Rings of Quotients of Group Rings

1969 ◽  
Vol 21 ◽  
pp. 865-875 ◽  
Author(s):  
W. D. Burgess
Keyword(s):  
Group Ring ◽  
Group Rings ◽  
Maximum Condition ◽  
Regular Group ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Completely Reducible ◽  
Rings Of Quotients ◽  
Prime Case

The group ring AG of a group G and a ring A is the ring of all formal sums Σg∈G agg with ag ∈ A and with only finitely many non-zero ag. Elements of A are assumed to commute with the elements of G. In (2), Connell characterized or completed the characterization of Artinian, completely reducible and (von Neumann) regular group rings ((2) also contains many other basic results). In (3, Appendix 3) Connell used a theorem of Passman (6) to characterize semi-prime group rings. Following in the spirit of these investigations, this paper deals with the complete ring of (right) quotients Q(AG) of the group ring AG. It is hoped that the methods used and the results given may be useful in characterizing group rings with maximum condition on right annihilators and complements, at least in the semi-prime case.

Extending the Property of a Maximal Right Ideal

2006 ◽  
Vol 13 (01) ◽  
pp. 163-172 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Dinh Van Huynh ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
The Self ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

We extend various properties from a direct summand X of a module M, whose complement is semisimple, to its trace in M or to M itself. The case when MR = RR and the properties are injectivity or P-injectivity is fully described. As applications, we extend some known results for right HI-rings and give a new characterization of semisimple rings. We conclude this paper by giving some conditions that yield the self-injectivity of von Neumann regular rings.

scholarly journals Onn-flat modules andn-Von Neumann regular rings

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Najib Mahdou
Keyword(s):  
Regular Ring ◽  
Local Rings ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Flat Modules ◽  
Von Neumann Regular ◽  
Regular Rings

We show that eachR-module isn-flat (resp., weaklyn-flat) if and only ifRis an(n,n−1)-ring (resp., a weakly(n,n−1)-ring). We also give a new characterization ofn-Von Neumann regular rings and a characterization of weakn-Von Neumann regular rings for (CH)-rings and for local rings. Finally, we show that in a class of principal rings and a class of local Gaussian rings, a weakn-Von Neumann regular ring is a (CH)-ring.

Properly algebraic and spectrum-finite ideals in Jordan systems

1993 ◽  
Vol 114 (1) ◽  
pp. 149-161 ◽  
Author(s):  
Ottmar Loos
Keyword(s):  
Triple System ◽  
Von Neumann ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Algebraic Elements ◽  
Algebraic Ideal ◽  
Jordan System

The two main results of this paper are:(i) The set of properly algebraic elements of a Jordan system (algebra, triple system or pair) over an uncountable field is an ideal.(ii) For a semiprimitive Banach Jordan system, the socle, the largest properly algebraic ideal, the largest properly spectrum-finite ideal and the largest von Neumann regular ideal all coincide.

A CLOSURE OPERATION IN RINGS

2001 ◽  
Vol 12 (07) ◽  
pp. 791-812
Author(s):  
PERE ARA ◽  
GERT K. PEDERSEN ◽  
FRANCESC PERERA
Keyword(s):  
Regular Element ◽  
Closure Operation ◽  
Von Neumann ◽  
Regular Elements ◽  
C Algebras ◽  
Partial Inverse ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Invertible Elements ◽  
Maximal Regular Ideal

We study the operation E → cl (E) defined on subsets E of a unital ring R, where x ∈ cl (E) if (x + Rb) ∩ E ≠ ∅ for each b in R such that Rx + Rb = R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl (L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl (Re) = Re + rad (R) if e is an idempotent in R, and cl (I) = I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad (R/I) = 0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an "open" subset Rpr in R, i.e. cl (R\Rpr) = R\Rpr. In fact, R\Rpr = cl (R\Rr), so that Rpr is the "algebraic interior" of the set Rr of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1 - xy)R(1 - yx) is contained in Rr. If I reg (R) denotes the maximal regular ideal in R and [Formula: see text] the set of quasi-invertible elements, defined and studied in [4], we prove that [Formula: see text]. Specializing to C*-algebras we prove that cl (E) coincides with the norm closure of E, when E is one of the five interesting sets R-1, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], and that Rpr coincides with the topological interior of Rr. We also show that the operation cl respects boundedness, self-adjointness and positivity.

scholarly journals On regular ideals in reduced rings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3715-3726 ◽  
Author(s):  
A.R. Aliabady ◽  
R. Mohamadian ◽  
S. Nazari
Keyword(s):  
Closed Subset ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Closed Set ◽  
Von Neumann ◽  
Arbitrary Ring ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Isolated Points ◽  
Maximal Regular Ideal

Let R be a commutative ring with identity and X be a Tychonoff space. An ideal I of R is Von Neumann regular (briefly, regular) if for every a ? I, there exists b ? R such that a = a2b. In the present paper, we obtain the general form of a regular ideal in C(X) which is OA, for some closed subset A of ?X, for which Ac?X ? (P(X))?, where P(X) is the set of all P-points of X. We show that the ideals and subrings such as CK(X), C?(X), C?(X), SocmC(X) and M?X\X are regular if and only if they are equal to the socle of C(X). We carry further the study of the maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them OPD-spaces) the maximal regular ideal is OX\I(X), where I(X) is the set of isolated points of X. Also, for this class, the socle of C(X) is the maximal regular ideal if and only if I(X) contains no infinite closed set. We also show that C(X) contains an ideal which is both essential and regular if and only if (P(X))? is dense in X. Finally it is shown that, for semiprimitive rings pure ideals are of the form OA which A is a closed subset of Max(R), also a P-point of X = Max(R) is introduced and it is shown that the maximal regular ideal of an arbitrary ring R is OX\P(X), which P(X) is the set of P-points of X = Max(R).

V-regular semigroups

1981 ◽  
Vol 88 (3-4) ◽  
pp. 275-291 ◽  
Author(s):  
K. S. S. Nambooripad ◽  
F. Pastijn
Keyword(s):  
Regular Semigroup ◽  
Transformation Semigroup ◽  
Inverse Semigroups ◽  
Full Transformation ◽  
Regular Semigroups ◽  
Von Neumann ◽  
Biordered Set ◽  
Von Neumann Regular ◽  
Strongly Regular

SynopsisA regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.

LEFT QUASI-MORPHIC RINGS

2008 ◽  
Vol 07 (06) ◽  
pp. 725-733 ◽  
Author(s):  
V. CAMILLO ◽  
W. K. NICHOLSON ◽  
Z. WANG
Keyword(s):  
Principal Ideal ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Finiteness Conditions ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Finite Sums ◽  
Left And Right ◽  
Left Annihilator

A ring R is called left quasi-morphic if, for each a ∈ R, there exist b and c in R such that Ra = l (b) and l (a) = Rc (where l (x) is the left annihilator). Every (von Neumann) regular ring is left quasi-morphic, as is every left morphic ring (b = c above). The main theorem of this paper is that, in a left quasi-morphic ring, finite intersections and finite sums of principal left ideals are again principal. This leads to structure theorems when mild finiteness conditions are imposed. In an earlier paper, the first two authors showed that left and right quasi-morphic rings have both these properties (on both sides), and used this to give a new characterization of the artinian principal ideal rings: They are just the left and right quasi-morphic rings with ACC on principal annihilators r (a), a ∈ R. Some extensions of this result are presented here.

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