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

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.

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Radicals in differential polynomial rings

International Journal of Algebra and Computation ◽

10.1142/s0218196721500041 ◽

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.

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On the f-Prime Radical of Near-Rings

Nearrings and Nearfields ◽

10.1007/1-4020-3391-5_16 ◽

2005 ◽

pp. 293-299

Author(s):

Satyanarayana Bhavanari ◽

Richard Wiegandtt

Keyword(s):

Prime Radical

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On the closure of the prime radical of a Banach algebra

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500018514 ◽

1993 ◽

Vol 36 (3) ◽

pp. 421-425 ◽

Cited By ~ 1

Author(s):

D. W. B. Somerset ◽

G. A. Willis

Keyword(s):

Banach Algebra ◽

Prime Radical ◽

Prime Ideals ◽

The Relationship

The relationship between the prime ideals and the primal ideals of a Banach algebra is investigated. It is shown that the closure of the prime radical of a Banach algebra may be properly contained in the intersection of the closed primal ideals of the algebra.

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ON SUPERNILPOTENT RADICALS WITH THE AMITSUR PROPERTY

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709000380 ◽

2009 ◽

Vol 80 (3) ◽

pp. 423-429 ◽

Cited By ~ 5

Author(s):

HALINA FRANCE-JACKSON

Keyword(s):

Jacobson Radical ◽

Semiprime Ring ◽

Nonzero Ideal ◽

Prime Radical ◽

Factor Ring ◽

Semisimple Class ◽

Primitive Ring

AbstractA radical α has the Amitsur property if α(A[x])=(α(A[x])∩A)[x] for all rings A. For rings R⊆S with the same unity, we call S a finite centralizing extension of R if there exist b1,b2,…,bt∈S such that S=b1R+b2R+⋯+btR and bir=rbi for all r∈R and i=1,2,…,t. A radical α is FCE-friendly if α(S)∩R⊆α(R) for any finite centralizing extension S of a ring R. We show that if α is a supernilpotent radical whose semisimple class contains the ring ℤ of all integers and α is FCE-friendly, then α has the Amitsur property. In this way the Amitsur property of many well-known radicals such as the prime radical, the Jacobson radical, the Brown–McCoy radical, the antisimple radical and the Behrens radical can be established. Moreover, applying this condition, we will show that the upper radical 𝒰(*k) generated by the essential cover *k of the class * of all *-rings has the Amitsur property and 𝒰(*k)(A[x])=𝒰(*k)(A)[x], where a semiprime ring R is called a *-ring if the factor ring R/I is prime radical for every nonzero ideal I of R. The importance of *-rings stems from the fact that a *-ring A is Jacobson semisimple if and only if A is a primitive ring.

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