ON SPECIAL RADICALS COINCIDING ON SIMPLE RINGS AND ON POLYNOMIAL RINGS

2003 ◽  
Vol 02 (01) ◽  
pp. 51-56 ◽  
Author(s):  
S. TUMURBAT
Keyword(s):  
Jacobson Radical ◽  
Polynomial Rings ◽  
Special Radical

Answering a problem of M. Ferrero, we construct a special radical δ such that δ is contained in the Jacobson radical J, δ and J coincides on simple rings and on polynomial rings, but δ ≠ J. For two special radicals with the above conditions, we give a criterion of their coincidence.

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

2014 ◽  
Vol 57 (3) ◽  
pp. 609-613 ◽  
Author(s):  
Alireza Nasr-Isfahani
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Base Ring ◽  
Necessary And Sufficient ◽  
Skew Polynomial

AbstractWe provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

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].

A Property Satisfying Reducedness over Centers

2021 ◽  
Vol 28 (03) ◽  
pp. 453-468
Author(s):  
Hailan Jin ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Finite Ring ◽  
Polynomial Rings ◽  
Free Algebras ◽  
Locally Finite ◽  
Nonzero Characteristic

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

scholarly journals Prime essential rings

1991 ◽  
Vol 34 (2) ◽  
pp. 241-250 ◽  
Author(s):  
B. J. Gardner ◽  
P. N. Stewart
Keyword(s):  
Prime Ideal ◽  
Continuous Functions ◽  
Polynomial Rings ◽  
Group Rings ◽  
Infinite Products ◽  
Rings Of Continuous Functions ◽  
Special Radical ◽  
Skew Group Rings ◽  
Infinite Sums ◽  
The Relationship

A ring R is prime essential if R is semiprime and for each prime ideal P of R, P ∩ I ≠0 whenever I is a nonzero two-sided ideal of R. Examples of prime essential rings include rings of continuous functions and infinite products modulo infinite sums. We show that the class of prime essential rings is closed under many familiar operations; in particular, we consider polynomial rings, matix rings, fixed rings and skew group rings. Also, we explore the relationship between prime essential rings and special radical classes, and we demonstrate how prime essential rings can be used to construct radical classes which are not special.

Special, radical, failure of reduction in psychiatry

2019 ◽  
Vol 42 ◽  
Author(s):  
Don Ross
Keyword(s):  
Mental Disorder ◽  
Brain Structure ◽  
Causal Structure ◽  
Network Models ◽  
Gambling Addiction ◽  
Brain Disorder ◽  
Mental Processes ◽  
Special Radical ◽  
Intentional Relationships

AbstractUse of network models to identify causal structure typically blocks reduction across the sciences. Entanglement of mental processes with environmental and intentional relationships, as Borsboom et al. argue, makes reduction of psychology to neuroscience particularly implausible. However, in psychiatry, a mental disorder can involve no brain disorder at all, even when the former crucially depends on aspects of brain structure. Gambling addiction constitutes an example.

Sign in / Sign up

Export Citation Format

Share Document