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].

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 THE COMPLEXITY OF THE EQUIVALENCE PROBLEM OVER FINITE RINGS

2011 ◽  
Vol 54 (1) ◽  
pp. 193-199 ◽  
Author(s):  
GÁBOR HORVÁTH
Keyword(s):  
Polynomial Time ◽  
Jacobson Radical ◽  
Equivalence Problem ◽  
Finite Ring ◽  
Polynomial Equations ◽  
Finite Rings

AbstractWe investigate the complexity of the equivalence problem over a finite ring when the input polynomials are written as sum of monomials. We prove that for a finite ring if the factor by the Jacobson radical can be lifted in the centre, then this problem can be solved in polynomial time. This result provides a step in proving a dichotomy conjecture of Lawrence and Willard (J. Lawrence and R. Willard, The complexity of solving polynomial equations over finite rings (manuscript, 1997)).

Radical Regularity in Differential Rings

1971 ◽  
Vol 23 (2) ◽  
pp. 197-201 ◽  
Author(s):  
Howard E. Gorman
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Prime Ideals ◽  
Differential Ring ◽  
Quotient Maps ◽  
The Stability ◽  
Definition Of ◽  
Finitely Generated Modules ◽  
Regular Rings ◽  
The Ideal

In [1], we discussed completions of differentially finitely generated modules over a differential ring R. It was necessary that the topology of the module be induced by a differential ideal of R and it was natural that this ideal be contained in J(R), the Jacobson radical of R. The ideal to be chosen, called Jd(R), was the intersection of those ideals which are maximal among the differential ideals of R. The question as to when Jd(R) ⊆ J(R) led to the definition of a class of rings called radically regular rings. These rings do satisfy the inclusion, and we showed in [1, Theorem 2] that R could always be “extended”, via localization, to a radically regular ring in such a way as to preserve all its differential prime ideals.In the present paper, we discuss the stability of radical regularity under quotient maps, localization, adjunction of a differential indeterminate, and integral extensions.

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].

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.

scholarly journals On group graded rings satisfying polynomial identities

1995 ◽  
Vol 37 (2) ◽  
pp. 205-210 ◽  
Author(s):  
A. V. Kelarev ◽  
J. Okniński
Keyword(s):  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Normal Band ◽  
Ring Theory ◽  
Semigroup Ring ◽  
Semigroup Algebras ◽  
Graded Rings ◽  
Finitely Generated ◽  
Polynomial Identities ◽  
Locally Finite

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18]). Several interesting results of this sort have appeared in the literature recently. In particular, it was proved in [1] that the Jacobson radical of every finitely generated PI-ring is nilpotent. For every commutative semigroup ring RS, it was shown in [11] that if J(R) is nil then J(RS) is nil. This result was generalized to all semigroup algebras satisfying polynomial identities in [15] (see [16, Chapter 21]). Further, it was proved in [12] that, for every normal band B, if J(R) is nilpotent, then J(RB) is nilpotent. A similar result for special band-graded rings was established in [13, Section 6]. Analogous theorems concerning nilpotency and local nilpotency were proved in [2] for rings graded by finite and locally finite semigroups.

On Group Rings

1970 ◽  
Vol 22 (2) ◽  
pp. 249-254 ◽  
Author(s):  
D. B. Coleman
Keyword(s):  
Nilpotent Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Characteristic Zero ◽  
Sylow Subgroup ◽  
Jacobson Radical ◽  
Closed Field ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced byThe first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

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