• Title, Summary, Keyword: commutative monoid

Search Result 11, Processing Time 0.026 seconds

COMMUTATIVE MONOID OF THE SET OF k-ISOMORPHISM CLASSES OF SIMPLE CLOSED k-SURFACES IN Z3

  • Han, Sang-Eon
    • Honam Mathematical Journal
    • /
    • v.32 no.1
    • /
    • pp.141-155
    • /
    • 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}.

SOME EXAMPLES OF QUASI-ARMENDARIZ RINGS

  • Hashemi, Ebrahim
    • Bulletin of the Korean Mathematical Society
    • /
    • v.44 no.3
    • /
    • pp.407-414
    • /
    • 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.

ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES

  • ROMANOWSKA, ANNA B.;SMITH, JONATHAN D.H.
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.5
    • /
    • pp.1587-1606
    • /
    • 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.

On spanning column rank of matrices over semirings

  • Song, Seok-Zun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.32 no.2
    • /
    • pp.337-342
    • /
    • 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.

  • PDF

Some Analogues of a Result of Vasconcelos

  • DOBBS, DAVID EARL;SHAPIRO, JAY ALLEN
    • Kyungpook Mathematical Journal
    • /
    • v.55 no.4
    • /
    • pp.817-826
    • /
    • 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.

Zero-divisors of Semigroup Modules

  • Nasehpour, Peyman
    • Kyungpook Mathematical Journal
    • /
    • v.51 no.1
    • /
    • pp.37-42
    • /
    • 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).

Weak Normality and Strong t-closedness of Generalized Power Series Rings

  • Kim, Hwan-Koo;Kwon, Eun-Ok;Kwon, Tae-In
    • Kyungpook Mathematical Journal
    • /
    • v.48 no.3
    • /
    • pp.443-455
    • /
    • 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.

GRADED PRIMAL SUBMODULES OF GRADED MODULES

  • Darani, Ahmad Yousefian
    • Journal of the Korean Mathematical Society
    • /
    • v.48 no.5
    • /
    • pp.927-938
    • /
    • 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.

ON t-ALMOST DEDEKIND GRADED DOMAINS

  • Chang, Gyu Whan;Oh, Dong Yeol
    • Bulletin of the Korean Mathematical Society
    • /
    • v.54 no.6
    • /
    • pp.1969-1980
    • /
    • 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.

PF-rings of Generalized Power Series

  • Kim, Hwankoo;Kwon, Tae In
    • Kyungpook Mathematical Journal
    • /
    • v.47 no.1
    • /
    • pp.127-132
    • /
    • 2007
  • In this paper, we show that if R is a commutative ring with identity and (S, ${\leq}$) is a strictly totally ordered monoid, then the ring [[$R^{S,{\leq}}$]] of generalized power series is a PF-ring if and only if for any two S-indexed subsets A and B of R such that $B{\subseteq}ann_R(|A)$, there exists $c{\in}ann_R(A)$ such that $bc=b$ for all $b{\in}B$, and that for a Noetherian ring R, $[[R^{S,{\leq}}$]] is a PP ring if and only if R is a PP ring.

  • PDF