• Korean Journal of Mathematics
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• v.19 no.4
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• pp.403-407
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• 2011

In this paper, we study Noetherian Boolean rings. We show that if R is a Noetherian Boolean ring, then R is finite and $R{\simeq}(\mathbb{Z}_2)^n$ for some integer $n{\geq}1$. If R is a Noetherian ring, then R/J is a Noetherian Boolean ring, where J is the intersection of all ideals I of R with |R/I| = 2. Thus R/J is finite, and hence the set of ideals I of R with |R/I| = 2 is finite. We also give a short proof of Hayes's result : For every polynomial $f(x){\in}\mathbb{Z}[x]$ of degree $n{\geq}1$, there are irreducible polynomials $g(x)$ and $h(x)$, each of degree $n$, such that $g(x)+h(x)=f(x)$.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.847-865
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• 2017

For a finite or an infinite set X, let $2^X$ be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set $2^X{\setminus}\{X,{\emptyset}\}$, with M adjacent to N if $M{\cap}N={\emptyset}$. In this paper, we characterize the annihilating-ideal graphs $\mathbb{AG}(R)$ that are blow-ups of strong Boolean graphs, complemented graphs and preatomic graphs respectively. In particular, for a commutative ring R such that AG(R) has a maximum clique S with $3{\leq}{\mid}V(S){\mid}{\leq}{\infty}$, we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a complemented graph, if and only if R is a reduced ring. If assume further that R is decomposable, then we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a blow-up of a pre-atomic graph. We also study the clique number and chromatic number of the graph $\mathbb{AG}(R)$.

• Kyungpook Mathematical Journal
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• v.48 no.2
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• pp.195-206
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• 2008

The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not of this kind. An interesting feature of the considerations involved here is the use made of nonmeasurable cardinals. In addition, the integer-valued function rings for Boolean frames are described in terms of internal lattice-ordered ring properties.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1179-1187
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• 2018

We study the class of rings R with the property that for $x{\in}R$ at least one of the elements x and 1 + x are tripotent. We prove that a commutative ring has this property if and only if it is a subring of a direct product $R_0{\times}R_1{\times}R_2$ such that $R_0/J(R_0){\cong}{\mathbb{z}}_2$, for every $x{\in}J(R_0)$ we have $x^2=2x$, $R_1$ is a Boolean ring, and $R_3$ is a subring of a direct product of copies of ${\mathbb{z}}_3$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.47-58
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• 2021

In this article, we completely determine the rings for which every element is a sum of a nilpotent and three tripotents that commute with one another. We discuss this property for some extensions of rings, including group rings.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.539-558
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• 2019

An ideal of a commutative ring is called a d-ideal if it contains the annihilator of the annihilator of each of its elements. Denote by DId(A) the lattice of d-ideals of a ring A. We prove that, as in the case of f-rings, DId(A) is an algebraic frame. Call a ring homomorphism "compatible" if it maps equally annihilated elements in its domain to equally annihilated elements in the codomain. Denote by $SdRng_c$ the category whose objects are rings in which the sum of two d-ideals is a d-ideal, and whose morphisms are compatible ring homomorphisms. We show that $DId:\;SdRng_c{\rightarrow}CohFrm$ is a functor (CohFrm is the category of coherent frames with coherent maps), and we construct a natural transformation $RId{\rightarrow}DId$, in a most natural way, where RId is the functor that sends a ring to its frame of radical ideals. We prove that a ring A is a Baer ring if and only if it belongs to the category $SdRng_c$ and DId(A) is isomorphic to the frame of ideals of the Boolean algebra of idempotents of A. We end by showing that the category $SdRng_c$ has finite products.

• Journal of the Korean Institute of Telematics and Electronics
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• v.24 no.6
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• pp.1040-1048
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• 1987

We propose a hardware architecture called Multiring which is applicable for various geometrical operations on rectilinear objects such as design rule checking in VLSI layout and many image processing operations including noise suppression and coutour extraction. It has both a fast execution speed and extremely high flexibility. The whole architecture is mainly divided into four parts` I/O between host and Multiring, ring memory, linear processor array and instruction decoder. Data transmission between host and Multiring is bit serial thereby reducing the bandwidth requirement for teh channel and the number of external pins, while each row data in the bit map stored in ring memory is processed in the corresponding processor in full parallelism. Each processor is simultaneously configured by the instruction decoder/controller to perform one of the 16 basic instructions such as Boolean (AND, OR, NOT, and Copy), geometrical(Expand and Shrink), and I/O operations each ring cycle, which gives Multiring maximal flexibility in terms of design rule change or the instruction set enhancement. Correct functional behavior of Multiring was confirmed by successfully running a software simulator having one-to-one structural correspondence to the Multiring hardware.