• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.253-266
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• 2014

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.169-177
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• 2003

Recently, K. Watanabe Showed that the Hilbert-Kunz multiplicity of a toric ring is a rational number. In this paper we give an explicit formula to compute the Hilbert-Kunz multiplicity of two-dimensional toric rings. This formula also shows that the Hilbert-Kunz multiplicity of a two-dimensional non-regular toric ring is at least 3/2.

• East Asian mathematical journal
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• v.16 no.1
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• pp.105-110
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• 2000

Baer rings were introduced by Kaplansky [3] to abstract various properties of rings of operators on a Hilbert spaces. On the other hand, p.p. rings were introduced by A. Hattori [2] to study the torsion theory. In this paper we introduce Baer near-rings and p.p. near-rings and study some properties and give some examples.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.265-270
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• 1995

Let $K_1, \ldots, K_n$ be fields of transcendence degrees $t_1, \ldots, t_n$ respectively over a common subfield F. O'Carroll and Qureshi [7] conjectured that the tensor product $R = K_1 \otimes K_2 \otimes \ldots \otimes K_n$ is an equidimensional Hilbert ring and proved the conjecture in special cases. Trung proved the conjecture [9] and O'Carroll, Bowman and Howie [3,5] generalized the Trung's result in two directions and obtained two theorems stated below.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.363-372
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• 2010

In this paper, we introduce and study the generalized inverse $A^{(1,2)}_{T,S}$ with the prescribed range T and null space S of an adjointable operator A from one Hilbert $C^*$-module to another, and get some analogous results known for finite matrices over the complex field or associated rings, and the Hilbert space operators.

• East Asian mathematical journal
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• v.4
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• pp.87-99
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• 1988

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.137-142
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• 2001

In this paper, we initiate the study of some annihilator conditions on polynomials which were used by Kaplansky [4] to abstract algebras of bounded linear operators on a Hilbert spaces with Baer condition. On the other hand, p.p. rings were introduced by A. Hattori [3] to study the torsion theory. The purpose of this paper is to introduce near-rings with Baer condition and near-rings with p.p. condition which are somewhat different from the ring case, and to extend a results of Armendarz [1] to polynomial near-rings with Baer condition in somewhat different way of Birkenmeier and Huang [2].

• The Pure and Applied Mathematics
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• v.10 no.3
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• pp.177-183
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• 2003

In this paper, we initiate the study of some annihilator conditions on polynomials which were used by Kaplansky [Rings of operators. W. A. Benjamin, Inc., New York, 1968] to abstract the algebra of bounded linear operators on a Hilbert spaces with Baer condition. On the other hand, p.p.-rings were introduced by Hattori [A foundation of torsion theory for modules over general rings. Nagoya Math. J. 17 (1960) 147-158] to study the torsion theory. The purpose of this paper is to introduce the near-rings with Baer condition and near-rings with p.p. condition which are somewhat different from ring case, and to extend a results of Armendariz [A note on extensions of Baer and P.P.-rings. J. Austral. Math. Soc. 18 (1974), 470-473] and Jøndrup [p.p. rings and finitely generated flat ideals. Proc. Amer. Math. Soc. 28 (1971) 431-435].

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1251-1258
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• 2020

This paper establishes a formula changing the ring from a Noetherian local ring A of dimension d > 0 containing the residue field k to the polynomial ring in d variables k[X₁, X₂, …, X_d] for mixed multiplicities. And as consequences, we get a formula for the multiplicity of Rees rings and formulas for mixed multiplicities and the multiplicity of Rees rings of quotient rings of A by highest dimensional associated prime ideals of A.

• Journal of applied mathematics & informatics
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• v.38 no.3_4
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• pp.271-276
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• 2020

Let R be a commutative ring with identity, R[X] the polynomial ring over R, ∗ a radical operation on R and ⋆ a radical operation of finite character on R[X]. In this paper, we give Hilbert basis theorem for rings with ∗-Noetherian spectrum. More precisely, we show that if (I^∗R[X])^⋆ = (IR[X])^⋆ and (I^∗R[X])^⋆ ∩ R = I^∗ for all ideals I of R, then R has ∗-Noetherian spectrum if and only if R[X] has ⋆-Noetherian spectrum. This is a generalization of a well-known fact that R has Noetherian spectrum if and only if R[X] has Noetherian spectrum.