scholarly journals On Sands' questions concerning strong and hereditary radicals

1986 ◽  
Vol 28 (1) ◽  
pp. 1-3 ◽  
Author(s):  
E. R. Puczyłowski
Keyword(s):  
Prime Radical ◽  
Hereditary Radical

[7] Sands raised the following questions:(1) Must a hereditary radical which is right strong be left strong?(2) Must a right hereditary radical be left hereditary?(3) (Example 6) Does there exist a right strong radical containing the prime radical β which is not left strong or hereditary?Negative answers to questions (1) and (2) were given by Beidar [1].In this paper we present different examples to answer (1) and (2), and we answer (3). We prove that the strongly prime radical defined in [4, 5] is right but not left strong. In the proof we use an example given by Parmenter, Passman and Stewart [6]. The same example and the strongly prime radical are used to answer (2) and (3).

ON EXTENSION OF PRIME RADICAL IN 2-PRIMAL NEAR-RINGS

2020 ◽  
Vol 9 (3) ◽  
pp. 1339-1348
Author(s):  
B. Elavarasan ◽  
K. Porselvi and J. Catherine Grace John ◽  
Porselvi J. Catherine Grace John
Keyword(s):  
Prime Radical

scholarly journals Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad ◽  
T. Srinivas
Keyword(s):  
Jacobson Radical ◽  
Paper Properties ◽  
Hereditary Radical ◽  
Constant Part ◽  
The Right

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

On the prime radical of a module over a commutative ring

1992 ◽  
Vol 20 (12) ◽  
pp. 3593-3602 ◽  
Author(s):  
James Jenkins ◽  
Patrick F. Smith
Keyword(s):  
Commutative Ring ◽  
Prime Radical

ON THE GROENEWALD-HEYMAN STRONGLY PRIME RADICAL

1984 ◽  
Vol 7 (3) ◽  
pp. 225-240 ◽  
Author(s):  
M. M. Parmenter ◽  
P. N. Stewart ◽  
R. Wiegandt
Keyword(s):  
Prime Radical

scholarly journals On the Prime Radical of a Hypergroupoid

2005 ◽  
Vol 1 (3) ◽  
pp. 234-238 ◽  
Author(s):  
Gursel Yesilot
Keyword(s):  
Prime Radical

A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

2007 ◽  
Vol 06 (02) ◽  
pp. 337-353 ◽  
Author(s):  
MAHMOOD BEHBOODI
Keyword(s):  
Prime Ring ◽  
Artinian Ring ◽  
Prime Radical ◽  
Nilpotent Elements ◽  
Noetherian Domain ◽  
Prime Submodules ◽  
Strongly Nilpotent ◽  
Commutative Domain ◽  
Torsion Modules

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.

Radicals in differential polynomial rings

2020 ◽  
pp. 1-17
Author(s):  
Jongwook Baeck ◽  
Nam Kyun Kim ◽  
Yang Lee
Keyword(s):  
Differential Polynomial ◽  
Prime Radical ◽  
Polynomial Rings

In this paper, we present new characterizations of several radicals of differential polynomial rings, including the Levitzki radical, strongly prime radical, and uniformly strongly prime radical in terms of the related [Formula: see text]-radical.

On the f-Prime Radical of Near-Rings

2005 ◽  
pp. 293-299
Author(s):  
Satyanarayana Bhavanari ◽  
Richard Wiegandtt
Keyword(s):  
Prime Radical
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