Subdirectly Irreducible DQC Rings

1971 ◽  
Vol 14 (4) ◽  
pp. 495-498 ◽  
Author(s):  
W. Burgess ◽  
M. Chacron
Keyword(s):  
Division Ring ◽  
Subdirectly Irreducible ◽  
Simple Ring

AbstractTwenty-five years ago McCoy published a characterization of commutative subdirectly irreducible rings. This result was generalized by Thierrin to duo rings with the word “field” which appeared in McCoy's theorem replaced by “division ring”. The purpose of this note is to give another generalization in which the words “division ring” will be replaced by “simple ring with 1 ”. The techniques resemble those of McCoy and Thierrin.

scholarly journals IRREDUCIBLE QUASIORDERS OF MONOUNARY ALGEBRAS

2012 ◽  
Vol 93 (3) ◽  
pp. 259-276 ◽  
Author(s):  
DANICA JAKUBÍKOVÁ-STUDENOVSKÁ ◽  
REINHARD PÖSCHEL ◽  
SÁNDOR RADELECZKI
Keyword(s):  
Partial Order ◽  
Unary Operation ◽  
Natural Order ◽  
Monounary Algebra ◽  
Natural Partial Order ◽  
Ordered Structure ◽  
Subdirectly Irreducible ◽  
Irreducible Elements ◽  
Algebraic Counterpart

AbstractRooted monounary algebras can be considered as an algebraic counterpart of directed rooted trees. We work towards a characterization of the lattice of compatible quasiorders by describing its join- and meet-irreducible elements. We introduce the limit $\cB _\infty $ of all $d$-dimensional Boolean cubes $\Two ^d$ as a monounary algebra; then the natural order on $\Two ^d$ is meet-irreducible. Our main result is that any completely meet-irreducible quasiorder of a rooted algebra is a homomorphic preimage of the natural partial order (or its inverse) of a suitable subalgebra of $\cB _\infty $. For a partial order, it is known that complete meet-irreducibility means that the corresponding partially ordered structure is subdirectly irreducible. For a rooted monounary algebra it is shown that this property implies that the unary operation has finitely many nontrivial kernel classes and its graph is a binary tree.

On a Theorem of Herstein

1969 ◽  
Vol 21 ◽  
pp. 1348-1353 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Division Ring ◽  
Monic Polynomial ◽  
Constant Term ◽  
Multiplicative Semigroup ◽  
Simple Ring ◽  
Left Hand ◽  
Ring Of Integers ◽  
Image Position

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

scholarly journals Completely simple endomorphism rings of modules

2018 ◽  
Vol 19 (2) ◽  
pp. 223
Author(s):  
Victor Bovdi ◽  
Mohamed Salim ◽  
Mihail Ursul
Keyword(s):  
Commutative Rings ◽  
Compact Ring ◽  
Endomorphism Rings ◽  
Locally Compact ◽  
Ring Topology ◽  
Simple Ring ◽  
Finite Topology ◽  
Completely Simple ◽  
The Ideal

<p>It is proved that if A<sub>p</sub> is a countable elementary abelian p-group, then: (i) The ring End (A<sub>p</sub>) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (A<sub>p</sub>)/I, where I is the ideal of End (A<sub>p</sub>) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (A<sub>p</sub>) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained.</p>

scholarly journals Green's Relations in Rings and Completely Simple Rings

2018 ◽  
Vol 14 (2) ◽  
pp. 7965-7974
Author(s):  
Florion Cela
Keyword(s):  
Division Ring ◽  
Green’S Relations ◽  
Simple Ring ◽  
Green's Relations ◽  
Completely Simple

In this paper we prove that which of Green's relations $\mathcal{L,R,H}$ and $\mathcal{D}$ in rings preserve the minimality of quasi-ideal. By this it is possible to show the structure of the classes generated by the above relations which have a minimal quasi ideal. For the completely simple rings we show that they are generated by the union of zero with a $\mathcal{D} $-class. Also we emphasize that a completely simple ring coincides with the union of zero with a $\mathcal{D} $-class if and only if it is a division ring.

scholarly journals A simple ring separating certain radicals

1975 ◽  
Vol 16 (1) ◽  
pp. 29-31 ◽  
Author(s):  
G. A. P. Heyman ◽  
W. G. Leavitt
Keyword(s):  
Homomorphic Image ◽  
Special Class ◽  
Radical Class ◽  
Subdirectly Irreducible ◽  
Prime Rings ◽  
Simple Ring ◽  
Special Radical ◽  
Hereditary Class

All rings considered will be associative. For a class M of rings let UM be the class of all rings having no non-zero homomorphic image in M. A hereditary class M of prime rings is called a “special class” [see 1, p. 191] if it has the property that when I ∈ M with I an ideal of a ring R, then R/I* ∈ Mwhere I* is the annihilator of I in R, and the corresponding radical class UM is then a “special radical”. Let S be the class of all subdirectly irreducible rings with simple heart.

Rings with Involution in which Every Trace is Nilpotent or Regular

1974 ◽  
Vol 26 (1) ◽  
pp. 130-137 ◽  
Author(s):  
Susan Montgomery
Keyword(s):  
Division Ring ◽  
Simple Ring ◽  
Direct Sum ◽  
Rings With Involution ◽  
Symmetric Element

A theorem of Marshall Osborn [15] states that a simple ring with involution of characteristic not 2 in which every non-zero symmetric element is invertible must be a division ring or the 2 × 2 matrices over a field. This result has been generalized in several directions. IfRis semi-simple and every symmetric element (or skew, or trace) is invertible or nilpotent, thenRmust be a division ring, the 2 × 2 matrices over a field, or the direct sum of a division ring and its opposite [6; 8; 13; 16].

scholarly journals On Duo Rings

1960 ◽  
Vol 3 (2) ◽  
pp. 167-172 ◽  
Author(s):  
G. Thierrin
Keyword(s):  
Commutative Rings ◽  
Prime Ideals ◽  
Subdirectly Irreducible ◽  
Nilpotent Elements

Following E. H. Feller [l], a ring R is called a duo ring if every one-sided ideal of R is a two-sided ideal.In the first part of this paper, we give some properties of duo rings and we show that the set of the nilpotent elements of a duo ring R is an ideal, the intersection of the completely prime ideals of R.It is easy to see that every duo ring is a subdirect sum of subdirectly irreducible duo rings. We give in the second part of this paper a characterization of the subdirectly irreducible duo rings. This characterization is quite similar to the characterization of the subdirectly irreducible commutative rings, due to N. H. McCoy [2], whose methods we use.

Isomorphisms of Simple Rings

1963 ◽  
Vol 15 ◽  
pp. 467-470
Author(s):  
Ti Yen
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Minimum Condition ◽  
Compatible Pair ◽  
Simple Ring ◽  
Common Extension ◽  
Necessary And Sufficient

Let A be a simple ring with minimum condition, and B1, B2, and C be regular subrings of A such that Bi > C, i = 1, 2. A pair of isomorphisms σi of Bi into A such that σi|C is the identity, and that Biσi are regular subrings of A (i = 1, 2), is called compatible if σ1|B1 ∩ B2 = σ2|B1 ∩ B2. Here σ|X means the restriction of σ to X. Bialynicki-Birula has proved some necessary and sufficient conditions that every compatible pair (σ1, σ2) has a common extention to an automorphism σ of A (1 ). When A is a division ring, he shows that the linear disjointness of the division subrings B1 and B2 is necessary and almost sufficient for the existence of a common extension of any compatible pair.

A CHARACTERIZATION OF SUBDIRECTLY IRREDUCIBLE ACTS

2015 ◽  
pp. 5-16 ◽  
Author(s):  
I. B. Kozhukhov ◽  
◽  
A. R. Haliullina ◽  
Keyword(s):  
Subdirectly Irreducible

scholarly journals A characterization of Fuzzy neighborhood commutative division rings

1993 ◽  
Vol 16 (4) ◽  
pp. 709-716 ◽  
Author(s):  
T. M. G. Ahsanullah ◽  
Fawzi Al-Thukair
Keyword(s):  
Fuzzy Set ◽  
Division Ring ◽  
Alternative Formulation ◽  
Division Rings

We give a characterization of fuzzy neighborhood commutative division ring; and present an alternative formulation of boundedness introduced in fuzzy neighborhood rings. The notion ofβ-restricted fuzzy set is considered.

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