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 Dynamical properties of profinite actions

2011 ◽  
Vol 32 (6) ◽  
pp. 1805-1835 ◽  
Author(s):  
MIKLÓS ABÉRT ◽  
GÁBOR ELEK
Keyword(s):  
Finite Index ◽  
Linear Groups ◽  
Regular Graphs ◽  
Weak Containment ◽  
Residually Finite Groups ◽  
The Family ◽  
Property T ◽  
Free Actions ◽  
Theoretical Consequence ◽  
Lattice Of Subgroups

AbstractWe study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

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 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].

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