Mostrando recursos 1 - 20 de 234

  1. Ulrich ideals of Gorenstein numerical semigroup rings with embedding dimension 3

    Numata, Takahiro
    The notion of Ulrich ideals was introduced by Goto et al.~\cite {GOTWY}. They developed an interesting theory on Ulrich ideals. In particular, they gave a characterization of Ulrich ideals of Gorenstein numerical semigroup rings that are generated by monomials. Using this result, in this paper, we investigate Ulrich ideals of Gorenstein numerical semigroup rings with embedding dimension~3 that are generated by monomials. In particular, we completely determine when such Ulrich ideals are existent in those rings.

  2. Integral domains in which every ideal is projectively equivalent to a prime ideal

    Lucas, Thomas G.; Mimouni, A.
    In this paper, we give complete characterizations of Noetherian domains and integrally closed domains in which every ideal is projectively equivalent to a prime ideal. We also characterize pullbacks satisfying this property and show how to construct integral domains in which every ideal is projectively equivalent to a prime ideal outside the context of Noetherian domains and integrally closed domains.

  3. On canonical modules of idealizations

    Loan, Nguyen Thi Hong
    Let $(R,\mathfrak {m})$ be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let $M$ be a finitely generated $R$-module. In this paper, we study the structure of the canonical module $K(R\ \mathbb {n}\ M)$ of the idealization $R\ \mathbb {n}\ M$ via the polynomial type introduced by Cuong~\cite {C}. In particular, we give a characterization for $K(R\ \mathbb {n}\ M)$ being Cohen-Macaulay and generalized Cohen-Macaulay.

  4. Arithmetical rank of strings and cycles

    Kimura, Kyouko; Mantero, Paolo
    Let $R$ be a polynomial ring over a field~$K$. To a given squarefree monomial ideal $I \subset R$, one can associate a hypergraph $\mathcal{H} (I)$. In this article, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $R/I$ when $\mathcal{H} (I)$ is a string or a cycle hypergraph.

  5. Bi-amalgamated algebras along ideals

    Kabbaj, S.; Louartiti, K.; Tamekkante, M.
    Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two commutative ring homomorphisms, and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \textit {bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B\times C$ given by \[ A\bowtie ^{f,g}(J,J'):=\big \{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big \}. \] In this paper, we investigate ring-theoretic properties of \textit {bi-amalgamations} and capitalize on previous work carried out on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and...

  6. Some remarks on biequidimensionality of topological spaces and Noetherian schemes

    Heinrich, Katharina
    There are many examples of the fact that dimension and codimension behave somewhat counterintuitively. In \cite {EGAIV1}, it is stated that a topological space is equidimensional, equicodimensional and catenary if and only if every maximal chain of irreducible closed subsets has the same length. We construct examples that show that this is not even true for the spectrum of a Noetherian ring. This gives rise to two notions of biequidimensionality, and we show how these relate to the dimension formula and the existence of a codimension function.

  7. Radical perfectness of prime ideals in certain integral domains

    Chang, Gyu Whan; Kim, Hwankoo
    For a UMT-domain $D$, we characterize when the polynomial ring $D[X]$ is $t$-compactly packed and every prime $t$-ideal of $D[X]$ is radically perfect. As a corollary, for a quasi-Pr\"ufer domain $D$, we also characterize when every prime ideal of $D[X]$ is radically perfect. Finally we introduce the concepts of Serre's conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively.

  8. A note on rational normal scrolls

    Barile, Margherita
    We give a general upper bound for the arithmetical rank of the ideals generated by the 2-minors of scroll matrices with entries in an arbitrary commutative unit ring.

  9. Direct summands of infinite-dimensional polynomial rings

    Asgharzadeh, Mohsen; Dorreh, Mehdi; Tousi, Massoud
    Let $k$ be a field and $R$ a pure subring of the infinite-dimensional polynomial ring $k[X_1,\ldots ]$. If $R$ is generated by monomials, then we show that the equality of height and grade holds for all ideals of~$R$. Also, we show $R$ satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by $k$-automorphism on $k[X_1,\ldots ]$. This provides several examples of non Noetherian Cohen-Macaulay rings (e.g., Veronese, determinantal and Grassmanian rings).

  10. Index

  11. Associated primes of local cohomology after adjoining indeterminates. Part 2: The general case

    Robbins, Hannah
    Let $A$ be a domain finitely generated as an algebra over a field, $k$ of characteristic zero, $ R=A[t_1,\dots ,t_\ell ]\quad \mbox {or}\quad A[[t_1,\dots ,t_\ell ]]$, and $I$ an ideal of $R$. If $A$ has a resolution of singularities, $Y_0$, which is the blowup of $A$ along an ideal of depth at least 2 and is covered by a finite number of open affines with $H^j(Y_0,\mathcal {O}_{Y_0})$ of finite length over $A$ for $j>0$, we prove that $\rm{Ass} _RH^i_I(R) $ is finite for every $i$. In particular, this holds when $A$ is a finite-dimensional normal domain with an isolated singularity...

  12. The Buchberger resolution

    Olteanu, Anda; Welker, Volkmar
    We define the Buchberger resolution, which is a graded free resolution of a monomial ideal in a polynomial ring. Its construction uses a generalization of the Buchberger graph and encodes much of the combinatorics of the Buchberger algorithm. The Buchberger resolution is a cellular resolution that, when it is minimal, coincides with the Scarf resolution. The simplicial complex underlying the Buchberger resolution is of interest for its own sake, and its combinatorics is not fully understood. We close with a conjecture on the clique complex of the Buchberger graph.

  13. Sperner property and finite-dimensional Gorenstein algebras associated to matroids

    Maeno, Toshiaki; Numata, Yasuhide
    We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gr\"obner fan of the defining ideal of our Gorenstein algebra.

  14. The cone of Betti tables over three non-collinear points in the plane

    Gheorghita, Iulia; Sam, Steven V
    We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.

  15. On a topological characterization of Prüfer $v$-multiplication domains among essential domains

    Finocchiaro, Carmelo Antonio; Tartarone, Francesca
    In this paper, we characterize the Pr\"ufer $v$-multiplication domain as a class of essential domains verifying an additional property on the closure of some families of prime ideals, with respect to the constructible topology.

  16. A Northcott type inequality for Buchsbaum-Rim coefficients

    Balakrishnan, R.; Jayanthan, A.V.
    In 1960, Northcott \cite {DGN} proved that, if $e_0(I)$ and $e_1(I)$ denote the 0th and first Hilbert-Samuel coefficients of an $\mathfrak m$-primary ideal $I$ in a Cohen-Macaulay local ring $(R,\mathfrak m)$, then $e_0(I)-e_1(I)\le \ell (R/I)$. In this article, we study an analogue of this inequality for Buchsbaum-Rim coefficients. We prove that, if $(R,\mathfrak m)$ is a two dimensional Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module contained in a free module $F$ with finite co-length, then $\rm{br} _0(M)-\rm{br} _1(M)\le \ell (F/M)$, where $\rm{br} _0(M)$ and $\rm{br} _1(M$) denote 0th and 1st Buchsbaum-Rim coefficients, respectively.

  17. Representations of finite posets over the ring of integers modulo a prime power

    Arnold, David; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru
    The classical category Rep$(S,\mathbb {Z}_{p})$ of representations of a finite poset $S$ over the field $\mathbb {Z}_{p}$ is extended to two categories, Rep$(S,\mathbb {Z}_{p^{m}})$ and uRep$(S,\mathbb {Z}_{p^{m}})$, of representations of $S$ over the ring $\mathbb {Z}_{p^{m}}$. A list of values of $S$ and $m$ for which Rep$(S,\mathbb {Z}_{p^{m}})$ or uRep$(S,\mathbb {Z}_{p^{m}})$ has infinite representation type is given for the case that $S$ is a forest. Applications include a computation of the representation type for certain classes of abelian groups, as the category of sincere representations in (uRep$(S,\mathbb {Z}_{p^{m}})$) Rep$(S,Z_{p^{m}})$ has the same representation type as (homocyclic) $(S,p^{m})$-groups, a class of...

  18. Multigraded modules of nested type

    Sabzrou, Hossein; Tousi, Massoud
    Extending the notion of monomial ideals of nested type, we introduce multigraded modules of nested type. We characterize them algebraically, resulting in the explicit construction of their dimension filtration. We compute several important invariants, including their Castelnuovo-Mumford regularity and projective dimension without {us\nobreak {ing}} the construction of their minimal graded free resolution. We show that they are pretty clean and hence sequentially Cohen-Macaulay.

  19. Pseudo-convergent sequences and Prüfer domains of integer-valued polynomials

    Loper, K. Alan; Werner, Nicholas J.
    Let $K$ be a field with rank one valuation and $V$ the valuation domain of $K$. For a subset $E$ of $V$, the ring of integer-valued polynomials on $E$ is \[ \Int (E, V) = \{f \in K[x] \mid f(E) \subseteq V \}. \] A question of interest regarding $\Int (E, V)$ is: for which $E$ is $\Int (E, V)$ a Pr\"{u}fer domain? In this paper, we contribute a partial answer to this question. We classify exactly when $\Int (E, V)$ is Pr\"{u}fer in the case where the elements of $E$ comprise a pseudo-convergent sequence in $V$. Our work expands...

  20. On the functoriality of marked families

    Lella, Paolo; Roggero, Margherita
    The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow associating to each ideal $J$ of this type a scheme $\MFScheme {J}$, called a $J$-marked scheme. In this paper, we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not...

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