Some questions for further research

Gilmer, Robert

doi:10.1007/978-0-387-36717-0_24

Cited by 46 publications

(83 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When D ⋆ = D, the semistar operation ⋆, restricted to F (D), is "the classical" star operation (cf. [15,Sections 32 and 34]). In this case, we will write that ⋆ is a (semi)star operation on D.…”

Section: Background and Preliminary Results On Semistar Operationsmentioning

confidence: 99%

See 1 more Smart Citation

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Baghdadi

Fontana

2004

Communications in Algebra

View full text Add to dashboard Cite

In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the "classical" concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural generalizations, to the semistar setting, of their corresponding "classical" concepts. As an application, among other results, we obtain a semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prüfer domains for Prüfer semistar multiplication domains.

show abstract

Section: Background and Preliminary Results On Semistar Operationsmentioning

confidence: 99%

“…[15], [22], [19]). Recently, new interest on these theories has been originated by the work by R. Matsuda and A. Okabe [30], where the notion of semistar operation was introduced, as a generalization of the notion of star operation.…”

Section: Introductionmentioning

confidence: 99%

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Baghdadi

Fontana

2004

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…The latter is performed via Gaussian elimination. Finally, it should be noted that the results of this paper hold more generally for any finite chain ring i.e., a ring in which all ideals are ordered by inclusion [3], [10]. Generalizations to finite commutative rings (see [9]) and finite abelian groups (as in [12, Sect.…”

Section: Discussionmentioning

confidence: 99%

“…As an example, for the system Σ = (Z + , R, B) with B = span {(3, 3, 3, · · · )} a kernel representation is given by (σ − 1)w = 0. Example 1.1: Consider Σ = (Z + , Z 9 , B) (i.e., p = 3; r = 2) with B = span{ (3,3,3 In contrast to what would hold in the field case, there does not exist a single polynomial a(ξ) ∈ R[ξ] such that B is given by a(σ)w = 0. The 1997 paper [1] introduces a specific type of kernel representation, called the "adapted form".…”

Section: Motivation and Problem Statementsmentioning

confidence: 99%

Row reduced representations of behaviors over finite rings

Kuijper

Pinto

Polderman

2007

2007 46th IEEE Conference on Decision and Control

View full text Add to dashboard Cite

Abstract-Row reduced representations of behaviors over fields posses a number of useful properties. Perhaps the most important feature is the predictable degree property. This property allows a finite parametrization of the module generated by the rows of the row reduced matrix with prior computable bounds. In this paper we study row-reducedness of representations of behaviors over rings of the form Zpr , where p is a prime number. Using a restricted calculus within Zpr we derive a meaningful and computable notion of row-reducedness. I. MOTIVATION AND PROBLEM STATEMENTSIn the behavioral theory, the central role is played by the set B of trajectories that characterize a dynamical system Σ, see the textbook [11]. In fact, a dynamical system is defined as a triple Σ = (T, W, B), where T is the time axis, W is the signal alphabet, and where B, the behavior of the system, is a subset of W T . In this paper we consider dynamical systems Σ = (Z + , R q , B), where R is the ring Z p r . Here p is a prime number and r is a positive integer. We study the theory of representations of these systems, in particular kernel representations, see also [7], [5]. For r ≥ 2 the ring Z p r is not a field. All multiples of p in Z p r are zero divisors and this induces several difficulties. Classical fundamental results for systems over a field are open problems for systems over the ring Z p r . One of these open problems is the development of a theory of row reduced representations and accompanying parametrization results. For behavioral systems over fields there exists a welldeveloped theory of representations, see e.g. [11], [14], [15], [16]. We define σ, the backward shift operator, acting on elements in W T as (σw)(k) = w(k + 1). Any behavior over a field that is linear, σ-invariant and complete (i.e., closed in the topology of point wise convergence) admits a kernel representation, that is, a representation of the form R(σ)w = 0, where R(ξ) is a polynomial matrix in the indeterminate ξ. As an example, for the system Σ = (Z + , R, B) with B = span {(3, 3, 3, · · · )} a kernel representation is given by (σ − 1)w = 0. For polynomial matrices over a field F, the concept of row reducedness is alternatively formulated in terms of the predictable-degree property (terminology from Forney's paper [2]), which is defined below. Recall that the row degree of a row polynomial vector is defined as the maximum of the degrees of its components. Definition 1.2: Let the matrix R(ξ) ∈ F g×q [ξ] with row degrees d 1 , . . . , d g . R(ξ) is said to have the predictabledegree property if for any nonzero polynomial vectorThus the row degree of a(ξ)R(ξ) can be predicted from the degrees in a(ξ) and the row degrees of R(ξ). For the field case it is proven in [2] and in [4, that the above property is equivalent to the property that the leading row coefficient matrix of R(ξ) has full row rank, i.e., that R(ξ) is row reduced. See also [13]. This provides an easy test to establish whether a kernel representation has the predictabledegree property or not. Fur...

show abstract

“…[9] and [11]) there appears the notion of an almost Dedekind domain and an almost Krull domain. An almost Dedekind (respectively almost Krull) domain R is defined as a commutative integral domain such that R P is a Dedekind domain (respectively Krull domain) for all non-zero prime ideals P of R. In particular, an almost Krull domain is almost Dedekind if and only if all non-zero prime ideals are maximal.…”

Section: Introductionmentioning

confidence: 99%

Almost Krull rings

Jespers

Wauters

1986

J Aust Math Soc A

View full text Add to dashboard Cite

The notion of an almost Krull domain is extended to rings satisfying a polynomial identity. Some general structural results are obtained. We also prove that skew polynomial rings R[X, a] remain almost Krull if R is an almost Krull ring. Finally, we study when semigroup rings R [S] are almost Krull rings, in the case when the group of quotients of S has the ascending chain condition on cyclic subgroups. An example is included to show that the general (semi-) group ring case is much more difficult to deal with.

show abstract

Some questions for further research

Cited by 46 publications

References 25 publications

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

Row reduced representations of behaviors over finite rings

Almost Krull rings

Contact Info

Product

Resources

About