Prime ideals and sheaf representation of a pseudo symmetric ring

Shin, Gooyong

doi:10.1090/s0002-9947-1973-0338058-9

Cited by 169 publications

(40 citation statements)

References 10 publications

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“…A prime ideal P of a ring R is called completely prime if R/P is a domain. Shin [19,Proposition 1.11] proved that a ring R is 2-primal if and only if every minimal prime ideal of R is completely prime. Lambek [13] called a ring R symmetric when rst = 0 implies rts = 0 for all r, s, t ∈ R, proving that a ring R is symmetric if and only if r 1 r 2 · · · r n = 0, with n any positive integer, implies r σ (1) r σ (2) · · · r σ (n) = 0 for any permutation σ of the set {1, 2, .…”

Section: Basic Structure Of Ni Ringsmentioning

confidence: 99%

“…For a 2-primal ring R Sun [21,Theorem 2.3] showed that R is pm if and only if Spec(R) is a normal space if and only if the maximal ideal spectrum of R is a retract of Spec(R), where Spec(R) is the prime spectrum of R. We apply the topological methods of Sun [21] and Shin [19] to analyze these conditions for NI rings, relating to the space of strongly prime ideals of R in place of Spec(R).…”

Section: Topological Conditions For Ni Ringsmentioning

confidence: 99%

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Structure and topological conditions of NI rings

2006

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We first construct an NI ring but not 2-primal from given any 2-primal ring, in a simpler way than well-known examples. We study the structure of NI rings relating to strongly prime ideals and show that minimal strongly prime ideals can be lifted in NI rings. A ring is called (respectively weakly) pm if every (respectively strongly) prime ideal is contained in a unique maximal ideal in it. For a 2-primal ring R Sun proved that R is pm if and only if Max(R) is a retract of Spec(R) if and only if Spec(R) is normal. In the present note we prove for an NI ring R that R is weakly pm if and only if Max(R) is a retract of SSpec(R) if and only if SSpec(R) is normal, where SSpec(R) is the space of strongly prime ideals of R. We also prove that R is weakly pm if and only if R is pm when R is a symmetric ring. We lastly consider several kinds of extensions of NI rings.

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Section: Basic Structure Of Ni Ringsmentioning

confidence: 99%

Section: Topological Conditions For Ni Ringsmentioning

confidence: 99%

Structure and topological conditions of NI rings

2006

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“…In recent years, a growing number of articles have studied the class of rings that is associated with the set N (R) of all nilpotent elements of a ring R . In 1973, Shin [21] proved that the prime radical P (R) coincides with the set of all nilpotent elements of R if and only if every minimal prime ideal is completely prime. In 1993, the term 2-primal, which satisfies P (R) = N (R) , was created by Birkenmeier et al [5].…”

Section: Introductionmentioning

confidence: 99%

Extensions and topological conditions of NJ rings

Jiang¹,

Wang²,

Ren³

2019

Turk J Math

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A ring R is said to be NJ if J(R) = N (R) . This paper mainly studies the relationship between NJ rings and related rings, and investigates the Dorroh extension, the Nagata extension, the Jordan extension, and some other extensions of NJ rings. At the same time, we also prove that if R is a weakly 2-primal α -compatible ring with an isomorphism α of R , then R[x; α] is NJ; if R is a weakly 2-primal δ -compatible ring with a derivation δ of R , then R[x; δ] is NJ. Moreover, we consider some topological conditions for NJ rings and show for a NJ ring R that R is J-pm if and only if J -Spec(R) is a normal space if and only if M ax(R) is a retract of J -Spec(R) .

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“…In this note, we find there are many researchers worked and attempted to find some results concerning semicommutative rings. G. Shin [57] showed for a ring R the following statements are equivalent: (i) R is semicommutative. (ii) For any a, b ∈ R, ab = 0 implies aRb = 0.…”

Section: Permuting N-semigeneralized Semiderivation Of (στ)-Semicommmentioning

confidence: 99%

New Types of Permuting n-Derivations with Their Applications on Associative Rings

Atteya

2019

Symmetry

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In this article, we introduce new generators of a permuting n-derivations to improve and increase the action of usual derivation. We produce a permuting n-generalized semiderivation, a permuting n-semigeneralized semiderivation, a permuting n-antisemigeneralized semiderivation and a permuting skew n-antisemigeneralized semiderivation of non-empty rings with their applications. Actually, we study the behaviour of those types and present their results of semiprime ring R. Examples of various results have also been included. That is, many of the branches of science such as business, engineering and quantum physics, which used a derivation, have the opportunity to invest them in solving their problems.

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Prime ideals and sheaf representation of a pseudo symmetric ring

Abstract: ABSTRACT. Almost symmetric rings and pseudo symmetric rings are introduced.The

Cited by 169 publications

References 10 publications

Structure and topological conditions of NI rings

Structure and topological conditions of NI rings

Extensions and topological conditions of NJ rings

New Types of Permuting n-Derivations with Their Applications on Associative Rings

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