scholarly journals Radical Classes Closed Under Products

2013 ◽  
Vol 21 (3) ◽  
pp. 103-132
Author(s):  
Barry Gardner
Keyword(s):  
Abelian Groups ◽  
Direct Products ◽  
Radical Theory ◽  
Associative Rings ◽  
Regular Rings

Abstract This is a survey of what is known about Kurosh-Amitsur radical classes which are closed under direct products. Associative rings, groups, abelian groups, abelian ℓ-groups and modules are treated. We have attempted to account for all published results relevant to this topic. Many or most of these were not, as published, expressed in radical theoretic terms, but have consequences for radical theory which we point out. A fruitful source of results and examples is the notion of slenderness for abelian groups together with its several variants for other structures. We also present a few new results, including examples and a demonstration that e-varieties of regular rings are product closed radical classes of associative rings.

COMMUTING PROPERTIES OF EXT

2013 ◽  
Vol 94 (2) ◽  
pp. 276-288 ◽  
Author(s):  
PHILL SCHULTZ
Keyword(s):  
Abelian Groups ◽  
Direct Products ◽  
Direct Sums

AbstractWe characterize the abelian groups $G$ for which the functors $\mathrm{Ext} (G, - )$ or $\mathrm{Ext} (- , G)$ commute with or invert certain direct sums or direct products.

scholarly journals ON -LIKE RADICALS OF RINGS

2012 ◽  
Vol 88 (2) ◽  
pp. 331-339 ◽  
Author(s):  
H. FRANCE-JACKSON ◽  
T. KHULAN ◽  
S. TUMURBAT
Keyword(s):  
Polynomial Ring ◽  
Open Problems ◽  
Radical Theory ◽  
Semisimple Ring ◽  
Associative Rings

AbstractLet$\alpha $be any radical of associative rings. A radical$\gamma $is called$\alpha $-like if, for every$\alpha $-semisimple ring$A$, the polynomial ring$A[x] $is$\gamma $-semisimple. In this paper we describe properties of$\alpha $-like radicals and show how they can be used to solve some open problems in radical theory.

scholarly journals Identities for existence varieties of regular semigroups

1989 ◽  
Vol 40 (1) ◽  
pp. 59-77 ◽  
Author(s):  
T.E. Hall
Keyword(s):  
Type Theorem ◽  
Inverse Semigroups ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Existence Variety ◽  
Completely Regular Semigroups ◽  
Locally Inverse Semigroups ◽  
Orthodox Semigroups ◽  
Regular Rings

A natural concept of variety for regular semigroups is introduced: an existence variety (or e-variety) of regular semigroups is a class of regular semigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively. Examples include the class of orthodox semigroups, the class of (regular) locally inverse semigroups and the class of regular E-solid semigroups. The lattice of e-varieties of regular semigroups includes the lattices of varieties of inverse semigroups and of completely regular semigroups. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities: such identities are then given for many e-varieties. The concept is meaningful in universal algebra, and as for regular semigroups could give interesting results for e-varieties of regular rings.

scholarly journals Hereditary radicals and bands of associative rings

1991 ◽  
Vol 51 (1) ◽  
pp. 62-72 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Structure Theory ◽  
Radical Theory ◽  
Associative Rings

AbstractBands of associative rings were introduced in 1973 by Weissglass. For the radicals playing most essential roles in the structure theory (in particular, for those of Jacobson, Baer, Levitsky, Koethe) it is shown how to find the radical of a band of rings. The technique of the general Kurosh-Amitsur radical theory is used to consider many radicals simultaneously.

Theories of modules closed under direct products

1992 ◽  
Vol 57 (2) ◽  
pp. 515-521
Author(s):  
Roger Villemaire
Keyword(s):  
Direct Product ◽  
Abelian Groups ◽  
Complete Theory ◽  
Direct Products ◽  
Horn Theory

AbstractWe generalize to theories of modules (complete or not) a result of U. Felgner stating that a complete theory of abelian groups is a Horn theory if and only if it is closed under products. To prove this we show that a reduced product of modules ΠFMi (i ϵ I) is elementarily equivalent to a direct product of ultraproducts of the modules Mi(i ϵ I).

scholarly journals Subalgebras, direct products and associated lattices of MV-algebras

1992 ◽  
Vol 34 (3) ◽  
pp. 301-307 ◽  
Author(s):  
L. P. Belluce ◽  
A. Di Nola ◽  
A. Lettieri
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Propositional Logic ◽  
Abelian Groups ◽  
Completeness Theorem ◽  
Direct Products ◽  
Algebraic Proof ◽  
Mv Algebras

MV-algebras were introduced by C. C. Chang [3] in 1958 in order to provide an algebraic proof for the completeness theorem of the Lukasiewicz infinite valued propositional logic. In recent years the scope of applications of MV-algebras has been extended to lattice-ordered abelian groups, AF C*-algebras [10] and fuzzy set theory [1].

scholarly journals Torsion in K0 of unit-regular rings

1995 ◽  
Vol 38 (2) ◽  
pp. 331-341 ◽  
Author(s):  
K. R. Goodearl
Keyword(s):  
Abelian Groups ◽  
Regular Algebras ◽  
Open Question ◽  
Regular Rings

We construct examples of unit-regular rings R for which K0(R) has torsion, thus answering a longstanding open question in the negative. In fact, arbitrary countable torsion abelian groups are embedded in K0 of simple unit-regular algebras over arbitrary countable fields. In contrast, in all these examples K0(R) is strictly unperforated.

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