scholarly journals Kurosh's chains of associative rings

1990 ◽  
Vol 32 (1) ◽  
pp. 67-69 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
E. R. Puczylowski
Keyword(s):  
Natural Number ◽  
Left Ideal ◽  
Homomorphic Image ◽  
Radical Class ◽  
Lower Radical ◽  
Closed Class ◽  
A Chain ◽  
Associative Rings

Let N be a homomorphically closed class of associative rings. Put N1 = Nl = N and, for ordinals a ≥ 2, define Nα (Nα) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in Nβ for some β<α. In this way we obtain a chain {Nα} ({Nα}), the union of which is equal to the lower radical class IN (lower left strong radical class IsN) determined by N. The chain {Nα} is called Kurosh's chain of N. Suliński, Anderson and Divinsky proved [7] that . Heinicke [3] constructed an example of N for which lN ≠ Nk for k = 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural number n ≥ 1 there exists a class N such that IN = Nn+l ≠ Nn. Some results concerning the termination of the chain {Nα} were obtained in [2,4]. In this paper we present some classes N with Nα = Nα for all α Using this and Beidar's example we prove that for every natural number n ≥ 1 there exists an N such that Nα = Nα for all α and Nn ≠ Nn+i = Nn+2. This in particular answers Question 6 of [4].

scholarly journals A generalisation of the lower radical class

1999 ◽  
Vol 59 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Robert McDougall
Keyword(s):  
Radical Class ◽  
Lower Radical ◽  
Closed Class ◽  
Associative Rings ◽  
Class Construction

In this work we demonstrate that the lower radical class construction on a homomorphically closed class of associative rings generates a radical class for any class of associative rings. We also give a new description of the upper radical class using the construction on an appropriate generating class.

A Note on Hypernilpotent Radical Properties for Associative Rings

1967 ◽  
Vol 19 ◽  
pp. 447-448 ◽  
Author(s):  
S. E. Dickson
Keyword(s):  
Homomorphic Image ◽  
Lower Radical ◽  
Closed Class ◽  
Radical Construction ◽  
Associative Rings

We work entirely in the category of associative rings. We show that if P1 is a homomorphically closed class which contains the zero rings, then the lower Kurosh radical P of P1 is the class P2 of all rings R such that every non-zero homomorphic image of R has non-zero ideals in P1, provided that P1 is closed under extensions by zero rings (i.e., if I is a P1-ideal of R and (R/I)2 = 0, then R ∈ P1). The latter assumption replaces the hypothesis that P1 be hereditary for ideals in a similar result of Anderson-Divinsky-Sulinsky in (2). This leads to a brief proof that the lower radical construction of Kurosh terminates at Pω0 (where ω0 is the first infinite ordinal) when P1 is a homomorphically closed class of associative rings containing the zero rings. This was proved for arbitrary homomorphically closed classes P1 of associative rings in (2).

A Note on Lower Radical Constructions for Associative Rings

1968 ◽  
Vol 11 (1) ◽  
pp. 23-30 ◽  
Author(s):  
A.G. Heinicke
Keyword(s):  
Radical Class ◽  
Lower Radical ◽  
Associative Rings

In [2], a construction for the lower radical class R∘ (η) with respect to a class η of rings was given as the union of an inductively defined ascending transfinite chain of classes of rings. It was shown there that this construction terminates, for associative rings, at ω∘, the first infinite ordinal, in the sense that if {ηα: α an ordinal} is the chain, then R∘ (η) =ηω∘. Also, examples of classes η for which R∘ (η) = η1, η2, η3 were given.

scholarly journals On the construction of the Kurosh lower radical class for associative rings

1972 ◽  
Vol 13 (3) ◽  
pp. 362-364 ◽  
Author(s):  
Daryl Kreiling ◽  
Terry L. Jenkins
Keyword(s):  
Radical Class ◽  
Lower Radical ◽  
Associative Rings

All rings considered are to be associative. For definitions not included in this paper see [2]. Let R be a ring and S a subring of R.

scholarly journals Some radical constructions for associative rings

1974 ◽  
Vol 18 (4) ◽  
pp. 442-446 ◽  
Author(s):  
B. J. Gardner
Keyword(s):  
Radical Class ◽  
Lower Radical ◽  
Abelian Categories ◽  
Associative Rings

Dickson's construction [1] of radical and semi-simple classes for certain abelian categories is a rather straightforward procedure in comparison with the methods traditionally used in more general situations. In §2 of the present paper we use a well-known characterization of the lower radical class to obtain, via consideration of maps with accessible images, a similar “homomorphic orthogonality” characterization of radical and semi-simple classes of associative rings. By substituting certain other subring properties for accessibility, we are then able to obtain simple constructions of various types of radical classes, including those which are strict in the sense first used by Kurosh [3] for groups.

Lower Radicals in Associative Rings

1969 ◽  
Vol 21 ◽  
pp. 466-476 ◽  
Author(s):  
J. F. Watters
Keyword(s):  
Associative Ring ◽  
Countable Number ◽  
Lower Radical ◽  
Closed Class ◽  
Radical Property ◽  
Image Position ◽  
Associative Rings

Given a homomorphically closed class of (not necessarily associative) rings , the lower radical property determined by is the least radical property for which all rings in are radical. Recently (7) a process of constructing the lower radical property from a class of associative rings has been given which terminates after a countable number of steps. In this process, an ascending chain of classesis obtained and the property of being a ring in the class is the lower radical property determined by . In Theorem 1 we give another characterization of the rings in the class , λ ∈ {1, 2, …, omega;0}, and a procedure for constructing the lower radical determined by in an arbitrary associative ring is given.

scholarly journals Lower radicals in nonassociative rings

1972 ◽  
Vol 14 (4) ◽  
pp. 419-423 ◽  
Author(s):  
R. Tangeman ◽  
D. Kreiling
Keyword(s):  
Simple Proof ◽  
Radical Class ◽  
Universal Class ◽  
Lower Radical ◽  
Hereditary Class ◽  
Radical Construction ◽  
Associative Rings

Let W be a universal class of (not necessarily associative) rings and let A ⊆ W. Kurosh has given in [6] a construction for LA, the lower radical class determined by A in W. Using this construction, Leavitt and Hoffmann have proved in [4] that if A is a hereditary class (if K ∈ A and I is an ideal of K, then I ∈ A), then LA is also hereditary. In this paper an alternate lower radical construction is given. As applications, a simple proof is given of the theorem of Leavitt and Hoffmann and a result of Yu-Lee Lee for alternative rings is extended to not necessarily associative rings.

scholarly journals Hereditary semisimple classes

1970 ◽  
Vol 11 (1) ◽  
pp. 7-8 ◽  
Author(s):  
W. G. Leavitt
Keyword(s):  
Homomorphic Image ◽  
The Other ◽  
Radical Class ◽  
Universal Class ◽  
Semisimple Class ◽  
Other Hand ◽  
Associative Rings

It is well-known (see e.g. [1, p. 5]) that a class ℳ of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:(a)if ℳ, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈ℳ.(b) If R∈ but R∉ℳ, then R has a non-zero ideal I∈, where ℳ = {K ∈ | every non-zero K/H∉ℳ}. In fact ℳ is the radical class whose semisimple class is ℳ. On the other hand, if ℘ is a radical class, then ℐ℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class ℳ is hereditary (that is, when R∈ℳ, then all its ideals are in ℳ), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2]). The condition (satisfied in all associative or alternative classes) is that ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].

scholarly journals The radical equation P(An) = (P(A))n

1975 ◽  
Vol 19 (3) ◽  
pp. 257-259 ◽  
Author(s):  
R. E. Propes
Keyword(s):  
Radical Class ◽  
Associative Rings

The purpose of this paper is to impose conditions on a radical class P so that the P-radical of the ring of n × n-matrices over a ring A is equal to the ring of n×n-matrices over the ring P(A). In (1), Amitsur gave such conditions, but with the stipulation that the radical class P contained all zero-rings (rings in which all products are zero). In what follows, we shall be working within the class of associative rings.

scholarly journals Essential ideals, homomorphically closed classes and their radicals

1982 ◽  
Vol 33 (3) ◽  
pp. 356-363 ◽  
Author(s):  
T. L. Jenkins ◽  
H. J. le Roux
Keyword(s):  
Homomorphic Image ◽  
Infinite Number ◽  
Radical Class ◽  
Essential Ideal ◽  
Class E

AbstractOlson and Jenkins defined E(M) to be the class of all rings each nonzero homomorphic image of which contains either a nonzero M-ideal or an essential ideal where M, is any class of rings. E(M) was proven to be a radical class and various classes M were considered. Here the class E(M) is partitioned into two classes: H the class of all rings each nonzero homomorphic image of which has a proper essential ideal and the class H(M) of all rings each nonzero homomorphic image of which contains an M-ideal. It is shown that H is a radical class and under certain conditions H(M) is also a radical class. Various properties placed on M yield several well-known radical classes and an infinite number of supernilpotent nonspecial radical classes is constructed.

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