• Honam Mathematical Journal
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• v.32 no.1
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• pp.141-155
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• 2010

In this paper we prove that with some hypothesis the set of k-isomorphism classes of simple closed k-surfaces in ${\mathbf{Z}}^3$ forms a commutative monoid with an operation derived from a digital connected sum, k ${\in}$ {18,26}. Besides, with some hypothesis the set of k-homotopy equivalence classes of closed k-surfaces in ${\mathbf{Z}}^3$ is also proved to be a commutative monoid with the above operation, k ${\in}$ {18,26}.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.407-414
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• 2007

In [12], McCoy proved that if R is a commutative ring, then whenever g(x) is a zero-divisor in R[x], there exists a nonzero c $\in$ R such that cg(x) = 0. In this paper, first we extend this result to monoid rings. Then for a monoid M, we give some examples of M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings. Every reduced ring is M-quasi-Armendariz for any unique product monoid M and any strictly totally ordered monoid $(M,\;{\leq})$. Also $T_4(R)$ is M-quasi-Armendariz when R is reduced and M-Armendariz.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.313-329
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• 2024

An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ_≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1587-1606
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• 2015

Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic $J{\acute{o}}nsson$-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.337-342
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• 1995

A semiring is a binary system $(S, +, \times)$ such that (S, +) is an Abelian monoid (identity 0), (S,x) is a monoid (identity 1), $\times$ distributes over +, 0 $\times s s \times 0 = 0$ for all s in S, and $1 \neq 0$. Usually S denotes the system and $\times$ is denoted by juxtaposition. If $(S,\times)$ is Abelian, then S is commutative. Thus all rings are semirings. Some examples of semirings which occur in combinatorics are Boolean algebra of subsets of a finite set (with addition being union and multiplication being intersection) and the nonnegative integers (with usual arithmetic). The concepts of matrix theory are defined over a semiring as over a field. Recently a number of authors have studied various problems of semiring matrix theory. In particular, Minc [4] has written an encyclopedic work on nonnegative matrices.

• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.927-938
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• 2011

Let G be an abelian monoid with identity e. Let R be a G-graded commutative ring, and M a graded R-module. In this paper we first introduce the concept of graded primal submodules of M an give some basic results concerning this class of submodules. Then we characterize the graded primal ideals of the idealization R(+)M.

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.37-42
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• 2011

Let M be an R-module and S a semigroup. Our goal is to discuss zero-divisors of the semigroup module M[S]. Particularly we show that if M is an R-module and S a commutative, cancellative and torsion-free monoid, then the R[S]-module M[S] has few zero-divisors of size n if and only if the R-module M has few zero-divisors of size n and Property (A).

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.817-826
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• 2015

Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property (${\star}$) if for each nonzero element $a{\in}R$, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (${\star}$), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property (${\star}$) if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property (${\star}$), but the converse is false.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1969-1980
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• 2017

Let ${\Gamma}$ be a nonzero torsionless commutative cancellative monoid with quotient group ${\langle}{\Gamma}{\rangle}$, $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}R_{\alpha}$ be a graded integral domain graded by ${\Gamma}$ such that $R_{{\alpha}}{\neq}\{0\}$ for all ${\alpha}{\in}{\Gamma},H$ be the set of nonzero homogeneous elements of R, C(f) be the ideal of R generated by the homogeneous components of $f{\in}R$, and $N(H)=\{f{\in}R{\mid}C(f)_v=R\}$. In this paper, we introduce the notion of graded t-almost Dedekind domains. We then show that R is a t-almost Dedekind domain if and only if R is a graded t-almost Dedekind domain and RH is a t-almost Dedekind domains. We also show that if $R=D[{\Gamma}]$ is the monoid domain of ${\Gamma}$ over an integral domain D, then R is a graded t-almost Dedekind domain if and only if D and ${\Gamma}$ are t-almost Dedekind, if and only if $R_{N(H)}$ is an almost Dedekind domain. In particular, if ${\langle}{\Gamma}{\rangle}$ isatisfies the ascending chain condition on its cyclic subgroups, then $R=D[{\Gamma}]$ is a t-almost Dedekind domain if and only if R is a graded t-almost Dedekind domain.

• Kyungpook Mathematical Journal
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• v.48 no.3
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• pp.443-455
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• 2008

For an extension $A\;{\subseteq}\;B$ of commutative rings, we present a sufficient conditio for the ring $[[A^{S,\;\leq}]]$ of generalized power series to be weakly normal (resp., stronglyt-closed) in $[[B^{S,\;\leq}]]$, where (S, $\leq$) be a torsion-free cancellative strictly ordered monoid. As a corollary, it can be applied to the ring of power series in infinitely many indeterminates as well as in finite indeterminates.