scholarly journals WHEN IS THE LOWER RADICAL DETERMINED BY A SET OF RINGS STRONG?

2004 ◽  
Vol 46 (2) ◽  
pp. 371-378 ◽  
Author(s):  
M. FILIPOWICZ ◽  
E. R. PUCZYOWSKI
Keyword(s):  
Lower Radical

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 a class of radicals of rings

1995 ◽  
Vol 59 (2) ◽  
pp. 184-192
Author(s):  
R. Mazurek
Keyword(s):  
Lower Radical ◽  
Lattice Of Subgroups

AbstractLet λ be a property that a lattice of submodules of a module may possess and which is preserved under taking sublattices and isomorphic images of such lattices and is satisfied by the lattice of subgroups of the group of integer numbers. For a ring R the lower radical Λ generated by the class λ(R) of R-modules whose lattice of submodules possesses the property λ is considered. This radical determines the unique ideal Λ (R) of R, called the λ-radical of R. We show that Λ is a Hoehnke radical of rings. Although generally Λ is not a Kurosh-Amitsur radical, it has the ADS-property and the class of Λ-radical rings is closed under extensions. We prove that Λ (Mn (R)) ⊆ Mn (Λ(R)) and give some illustrative examples.

scholarly journals On prime essential rings

1993 ◽  
Vol 47 (2) ◽  
pp. 287-290 ◽  
Author(s):  
Halina France-Jackson
Keyword(s):  
Prime Ideal ◽  
Nonzero Ideal ◽  
Lower Radical ◽  
Semisimple Class ◽  
Open Questions ◽  
Nonzero Intersection

A ring A is prime essential if A is semiprime and every prime ideal of A has a nonzero intersection with each nonzero ideal of A. We prove that any radical (other than the Baer's lower radical) whose semisimple class contains all prime essential rings is not special. This yields non-speciality of certain known radicals and answers some open questions.

scholarly journals On the mutagenic radical property

1984 ◽  
Vol 27 (3) ◽  
pp. 333-336
Author(s):  
G. Tzintzis
Keyword(s):  
Lower Radical ◽  
Radical Property

In their paper N. Divinsky and A. Sulinski [6] have introduced the notion of mutagenic radical property—that is, a radical property which is far removed from hereditariness—and constructed two such examples. The first is the lower radical property determined by a ring Swo (N. Divinsky [5]) and is an almost subidempotent radical property in the sense of F. Szász [9], and the second is a weakly supernilpotent radical property, that is the lower radical property determined by Swo and all nilpotent rings.

scholarly journals Radicals Of Polynomial Rings

1956 ◽  
Vol 8 ◽  
pp. 355-361 ◽  
Author(s):  
S. A. Amitsur
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Present Note ◽  
Polynomial Rings ◽  
Lower Radical

Introduction. Let R be a ring and let R[x] be the ring of all polynomials in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R[x]) and L(R[x]). The Jacobson radical J(R[x]) is shown to be a polynomial ring N[x] over a nil ideal N of R and the lower radical L(R[x]) is the polynomial ring L(R)[x].

The Hereditary Property in the Lower Radical Construction

1968 ◽  
Vol 20 ◽  
pp. 474-476 ◽  
Author(s):  
E. P. Armendariz ◽  
W. G. Leavitt
Keyword(s):  
Hereditary Property ◽  
Lower Radical ◽  
The Third ◽  
Closed Class ◽  
Radical Construction

All rings considered are associative. We show that if a homomorphically closed class P1 of rings is hereditary in the sense that every ideal of a ring in P1 is also in P1, then the lower Kurosh radical construction terminates at P3. This is an improvement on the result of Anderson, Divinsky, and Sulinski (3) showing that the lower radical construction terminates at P2 provided P1 is homomorphically closed, hereditary, and contains all zero rings. Examples are given to show that the third step is actually attained in some constructions.

scholarly journals On the lower radical construction of Tangeman and Kreiling

1990 ◽  
Vol 33 (2) ◽  
pp. 203-205
Author(s):  
A. D. Sands
Keyword(s):  
Lower Radical ◽  
Radical Construction

It is shown that the lower radical construction of Tangeman and Kreiling need not terminate at any ordinal.

scholarly journals The lower radical construction for non-associative rings: Examples and counterexamples

1979 ◽  
Vol 20 (2) ◽  
pp. 259-271
Author(s):  
B.J. Gardner
Keyword(s):  
Sufficient Conditions ◽  
Variety Of Algebras ◽  
Lower Radical ◽  
Radical Construction ◽  
Associative Rings

Some sufficient conditions are presented for the lower radical construction in a variety of algebras to terminate at the step corresponding to the first infinite ordinal. An example is also presented, in a variety satisfying some non-trivial identities, of a lower radical construction terminating in four steps.

Lower Radical Properties for Associative and Alternative Rings

1966 ◽  
Vol s1-41 (1) ◽  
pp. 417-424 ◽  
Author(s):  
A. Sulinski ◽  
R. Anderson ◽  
N. Divinsky
Keyword(s):  
Lower Radical

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].

Sign in / Sign up

Export Citation Format

Share Document