• The Pure and Applied Mathematics
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• v.2 no.1
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• pp.61-65
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• 1995

In this paper we introduce the notion of Artinian and obtain some properties of Artinian and Noetherian BCK-algebras.

• Journal of the Chungcheong Mathematical Society
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• v.8 no.1
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• pp.173-181
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• 1995

Let H and G be finite dimensional semisimple Hopf algebras and let A and B be left H and G-module algebras respectively. We use smash product algebras to show that 1) if A is right Artinian then $A^H$ is right Artinian, 2) $Soc\;V_A{\subset}Soc\;V_{A^H}$ and rad $V_A{\supset}\;radV_{A^H}$, 3) $K\;dim\;_BV_A=K\;dim\;_{B^G}V_{A^H}$.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.513-518
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• 2007

There is a one to one correspondence between Artinian algebras $k[x_1,...,x_n]/Ann(M)$ and finitely generated $k[x_1,...,x_n]-submodules$ M of $k[y_1,...,y_n]$ by Inverse System. In particular, any Artinian level algebra $k[x_1,...,x_n]/Ann(M)$ can be obtained when M is finitely generated by only maximal degree generators. We prove that H = (1, 4, 8, 13,..., 27, 8, 2) is not a level Artinian O-sequence using this inverse system.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.317-326
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• 2003

We classify all possible level Artinian O-sequences of socle degree 5 and type 3. Moreover, we show how to construct level Artinian algebras with those Hilbert functions using the sum of two ideals of finite sets of points in $P^2$ such that the ideal of the union of two sets is level.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.605-614
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• 2005

We show that an Artinian O-sequence $h_0,h_1,{\cdots},h_{d-1},h_d\;=\;h_{d-1},h_{d+l}\;>\;h_d$ of codimension 3 is not level when $h_{d-1}\;=\;h_d\;=\;d + i\;and\;h{d+1}\;=\;d+(i+1)\;for\;i\;=\;1,\;2,\;and\;3$, which is a partial answer to the question in [9]. We also introduce an algorithm for finding noncancelable Betti numbers of minimal free resolutions of all possible Artinian O-sequences based on the theorem of Froberg and Laksov in [2].

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.225-232
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• 1998

Let H be a finite dimensional Hopf algebra over a field k, and A be an H-module algebra over k which the H-action on A is D-continuous. We show that $Q_{max}(A)$, the maximal ring or quotients of A, is an H-module algebra. This is used to prove that if H is a finite dimensional semisimple Hopf algebra and A is a semiprime right(left) Goldie algebra than $A#H$ is a semiprime right(left) Goldie algebra. Assume that Asi a semiprime H-module algebra Then $A^H$ is left Artinian if and only if A is left Artinian.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.405-417
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• 2012

We find the Hilbert function and the minimal free resolution of a star-configuration in $\mathbb{P}^n$. The conditions are provided under which the Hilbert function of a star-configuration in $\mathbb{P}^2$ is generic or non-generic We also prove that if $\mathbb{X}$ and $\mathbb{Y}$ are linear star-configurations in $\mathbb{P}^2$ of types t and s, respectively, with $s{\geq}t{\geq}3$, then the Artinian k-algebra $R/(I_{\mathbb{X}}+I_{\mathbb{Y})$ has the weak Lefschetz property.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.443-450
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• 2011

In this paper, we construct a Gorenstein Artinian algebra R/J with non-unimodal Hilbert function h = (1, 13, 12, 13, 1) to investigate the algebraic structure of the ideal J in a polynomial ring R. For this purpose, we use a software system Macaulay 2, which is devoted to supporting research in algebraic geometry and commutative algebra.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.593-601
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• 1996

W. Liu [11] has studied fuzzy ideals of a ring, and many researchers [5,6,7,16] are engaged in extending the concepts. The notion of fuzzy ideals and its properties were applied to various areas: semigroups [8,9,10,13,15], distributive lattices [2], artinian rings [12], BCK-algebras [14], near-rings [1]. In this paper we obtained an exact analogue of fuzzy ideals for near-ring which was discussed in [5, 11].