• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.1-6
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• 1994

In this paper, some properties for a PI-ring satisfying the descending chain condition on essential left ideals are studied: Let R be a ring with a polynomial identity satisfying the descending chain condition on essential ideals. Then all minimal prime ideals in R are maximal ideals. Moreover, if R has only finitely many minimal prime ideals, then R is left and right Artinian. Consequently, if every primeideal of R is finitely generated as a left ideal, then R is left and right Artinian. A finitely generated PI-algebra over a commutative Noetherian ring satisfying the descending chain condition on essential left ideals is a finite module over its center.(omitted)

• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.1-19
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• 2000

We investigate in this paper the maximality of prime ideals in rings whose simple singular left R-modules are p-injective.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.781-786
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• 2013

Let * be a star-operation on an integral domain R. We show that if R is a $*_{\omega}$-Noetherian domain, then quasi-$*_{\omega}$-invertible prime $*_{\omega}$-ideals of R are minimal, and prime ideals of R minimal over a $*_{\omega}$-invertible $*_{\omega}$-ideal are minimal.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.9 no.2
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• pp.110-114
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• 2009

We introduce the concepts of intuitionistic fuzzy k-ideals and intuitionistic fuzzy prime k-ideals of a semiring. And we investigate some properties of them.

• Journal of the Korean Institute of Intelligent Systems
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• v.9 no.5
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• pp.509-512
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• 1999

We study semiprime fuzzy ideals semiprimary fuzzy ideals and their properties. We investigate that if a fuzzy ideal is semiprime and semiprimary then it is prime.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.987-1030
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• 2001

Some discrimination of modules whose endomorhism rings are prime is introduced, by means of structures of submodules inducing prime ideals of the endomorphism ring End(sub)R (M) of a left R-module (sub)RM over a ring R. Modules with non-prime endomorphism rings are contrapositively studied as well.

• Journal of the Korean Mathematical Society
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• v.60 no.2
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• pp.407-464
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• 2023

Let R be a commutative ring with identity. The structure theorem says that R is a PIR (resp., UFR, general ZPI-ring, π-ring) if and only if R is a finite direct product of PIDs (resp., UFDs, Dedekind domains, π-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations v or t as follows: An integral domain R is a Krull domain if and only if every nonzero proper principal ideal of R can be written as a finite v- or t-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation u on R, so that R is a general Krull ring if and only if every proper principal ideal of R can be written as a finite u-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

• Honam Mathematical Journal
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• v.28 no.3
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• pp.327-332
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• 2006

Prime elements and ${\bigwedge}$-irreducible elements are introduced, and related properties are investigated.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.1-4
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• 2001

In [2, 7.3.2], the set of attached prime ideals of local cohomology module $H_m^n(M)$ were calculated, where (A, m) be Noetherian local ring, M finite A-module and $dim_A(M)=n$, and also in the special case in which furthermore A is a homomorphic image of a Gornestien local ring (A', m') (see [2, 11.3.6]). In this paper, we shall obtain this set, by another way in this special case.

• Korean Journal of Mathematics
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• v.28 no.3
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• pp.603-611
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• 2020

In this manuscript, we discuss the relationship between prime rings and automorphisms satisfying differential identities involving commutators and anti-commutators on Lie ideals. In addition, we provide an example which shows that we cannot expect the same conclusion in case of semiprime rings.