Embedding Right Chain Rings in Chain Rings

1978 ◽  
Vol 30 (5) ◽  
pp. 1079-1086 ◽  
Author(s):  
H. H. Brungs ◽  
G. Törner
Keyword(s):  
Maximal Ideal ◽  
Coordinate Ring ◽  
Hjelmslev Plane ◽  
Chain Ring ◽  
Projective Hjelmslev Plane ◽  
Algebraic Version ◽  
Zero Divisors ◽  
Starting Point ◽  
Affine Hjelmslev Plane ◽  
A Chain

The following problem was the starting point for this investigation: Can every desarguesian affine Hjelmslev plane be embedded into a desarguesian projective Hjelmslev plane [8]? An affine Hjelmslev plane is called desarguesian if it can be coordinatized by a right chain ring R with a maximal ideal J(R) consisting of two-sided zero divisors. A projective Hjemslev plane is called desarguesian if the coordinate ring is in addition a left chain ring, i.e. a chain ring. This leads to the algebraic version of the above problem, namely the embedding of right chain rings into suitable chain rings. We can prove the following result.

Dual Numbers and Topological Hjelmslev Planes

1983 ◽  
Vol 26 (3) ◽  
pp. 297-302 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Dual Numbers ◽  
Hjelmslev Plane ◽  
Product Topology ◽  
Topological Dimension ◽  
Locally Compact ◽  
Projective Hjelmslev Plane ◽  
Zero Divisors ◽  
Unique Maximal Ideal

AbstractIn 1929 J. Hjelmslev introduced a geometry over the dual numbers ℝ+tℝ with t2 = Q. The dual numbers form a Hjelmslev ring, that is a local ring whose (unique) maximal ideal is equal to the set of 2 sided zero divisors and whose ideals are totally ordered by inclusion. This paper first shows that if we endow the dual numbers with the product topology of ℝ2, then we obtain the only locally compact connected hausdorfT topological Hjelmslev ring of topological dimension two. From this fact we establish that Hjelmslev's original geometry, suitably topologized, is the only locally compact connected hausdorfr topological desarguesian projective Hjelmslev plane to topological dimension four.

scholarly journals Ordered Desarguesian Affine Hjelmslev Planes

1978 ◽  
Vol 21 (2) ◽  
pp. 229-235 ◽  
Author(s):  
L. A. Thomas
Keyword(s):  
Local Ring ◽  
Division Ring ◽  
Affine Plane ◽  
Hjelmslev Plane ◽  
Zero Divisors ◽  
Affine Hjelmslev Plane ◽  
Desarguesian Affine Plane

A Desarguesian affine Hjelmslev plane (D.A.H. plane) may be coordinatized by an affine Hjelmslev ring (A.H. ring), which is a local ring whose radical is equal to the set of two-sided zero divisors and whose principal right ideals are totally ordered (cf. [3]). In his paper on ordered geometries [4], P. Scherk discussed the equivalence of an ordering of a Desarguesian affine plane with an ordering of its coordinatizing division ring. We shall define an ordered D.A.H. plane and follow Scherk's methods to extend his results to D.A.H. planes and their A.H. rings i.e., we shall show that a D.A.H. plane is ordered if and only if its A.H. ring is ordered. We shall also give an example of an ordered A.H. ring. Finally, we shall discuss some infinitesimal aspects of the radical of an ordered A.H. ring.

Through the Lens of the Adam Narrative: A Re-consideration of Sūrat al-Baqara

2015 ◽  
Vol 17 (2) ◽  
pp. 24-46 ◽  
Author(s):  
M.O. Klar
Keyword(s):  
Present Article ◽  
Dominant Theme ◽  
Starting Point ◽  
Lexical Items ◽  
A Chain

The thesis of a single pillar or axis around which the longer Medinan suras are structured has been highly influential in the field of sura unity, and scholarship on the structure and coherence of Sūrat al-Baqara has tended to work towards charting the progress of a dominant theme throughout the textual blocks that make up the sura. In order to achieve this, scholars have divided the sura into discrete blocks; many have posited a chain of lexical and thematic links from one block to the next; some have concentrated solely on the hinges and borders between these suggested textual blocks. The present article argues that such methods, while often in themselves illuminating, are by their very nature reductive. As such they can result in the oversight of important elements of the sura. From a starting point of the Adam pericope provided in Q. 2:30–9, this study will focus on the recurrence of a number of its lexical items throughout Sūrat al-Baqara. By methodically tracing the passage of repeated, loosely Fall-related, vocabulary, it will attempt to widen the contextual lens through which the sura's textual blocks are viewed, and establish a broader perspective on its coherence. Via a discussion of the themes of ‘gardens’, ‘parable’, ‘prostration’, ‘covenant’, ‘wrongdoing’ and finally ‘blindness’, this article will posit ‘garments’, not as a structural pillar, but as a pivot around which many of the repeated lexical items of the sura rotate.

Intermolecular Effects: I. The Constrained-Junction Model

1997 ◽  
Author(s):  
Burak Erman ◽  
James E. Mark
Keyword(s):  
Experimental Data ◽  
Free Energy ◽  
Tensile Strain ◽  
Theoretical Models ◽  
Rigorous Derivation ◽  
Starting Point ◽  
Link Model ◽  
A Chain ◽  
Network Modulus ◽  
Degree Of Entanglement

The classical theories of rubber elasticity presented in chapter 2 are based on a hypothetical chain which may pass freely through its neighbors as well as through itself. In a real chain, however, the volume of a segment is excluded to other segments belonging either to the same chain or to others in the network. Consequently, the uncrossability of chain contours by those occupying the same volume becomes an important factor. This chapter and the following one describe theoretical models treating departures from phantom-like behavior arising from the effect of entanglements, which result from this uncrossability of network chains. The chains in the un-cross-linked bulk polymer are highly entangled. These entanglements are permanently fixed once the chains are joined during formation of the network. The degree of entanglement, or degree of interpenetration, in a network is proportional to the number of chains sharing the volume occupied by a given chain. This is quite important, since the observed differences between experimental results on real networks and predictions of the phantom network theory may frequently be attributed to the effects of entanglements. The decrease in network modulus with increasing tensile strain or swelling is the best-known effect arising from deformation-dependent contributions from entanglements. The constrained-junction model presented in this chapter and the slip-link model presented in chapter 4 are both based on the postulate that, upon stretching, the space available to a chain along the direction of stretch is increased, thus resulting in an increase in the freedom of the chain to fluctuate. Similarly, swelling with a suitable diluent separates the chains from one another, decreasing their correlations with neighboring chains. Experimental data presented in figure 3.1 show that the modulus of a network does indeed decrease with both swelling and elongation, finally becoming independent of deformation, as should be the case for the modulus of a phantom network. Rigorous derivation of the modulus of a network from the elastic free energy for this case will be given in chapter 5. The starting point of the constrained-junction model presented in this chapter is the elastic free energy.

Locally Compact Hjelmslev Planes and Rings

1981 ◽  
Vol 33 (4) ◽  
pp. 988-1021 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Projective Planes ◽  
Equivalence Relations ◽  
Topological Spaces ◽  
Hjelmslev Plane ◽  
Projective Hjelmslev Plane ◽  
Quotient Spaces ◽  
Continuous Maps ◽  
Topological Plane ◽  
Line Point ◽  
Canonical Image

Affine and projective Hjelmslev planes are generalizations of ordinary affine and projective planes where two points (lines) may be joined by (may intersect in) more than one line (point). The elements involved in multiple joinings or intersections are neighbours, and the neighbour relations on points respectively lines are equivalence relations whose quotient spaces define an ordinary affine or projective plane called the canonical image of the Hjelmslev plane. An affine or projective Hjelmslev plane is a topological plane (briefly a TH-plane and specifically a TAH-plane respectively a TPH-plane) if its point and line sets are topological spaces so that the joining of non-neighbouring points, the intersection of non-neighbouring lines and (in the affine case) parallelism are continuous maps, and the neighbour relations are closed.In this paper we continue our investigation of such planes initiated by the author in [38] and [39].

Bioethics Testimony: Untangling the Strands and Testing Their Reliability

2005 ◽  
Vol 33 (2) ◽  
pp. 222-233 ◽  
Author(s):  
Bethany J. Spielman
Keyword(s):  
Moral Reasoning ◽  
Normative Ethics ◽  
Starting Point ◽  
A Chain

In The Abuse of Casuistry Jonsen and Toulmin describe one view of moral reasoning as follows:Those who take a rhetorical view of moral reasoning… do not assume that moral reasoning relies for its force on single chains of unbreakable deductions which link present cases back to some common starting point. Rather (they believe), this strength comes from accumulating many parallel, complementary considerations, which have to do with the current circumstances of the human individuals and communities involved and lend strength to our conclusions, not like links to a chain but like strands to a rope or roots to a tree.Whether or not all moral reasoning resembles “strands to a rope,” bioethics testimony certainly does. Bioethics testimony is eclectic, a composite of many loosely woven strands. Rarely, if ever, is bioethics testimony “a chain of unbreakable deductions.” Rarely is it “pure” ethics, much less pure normative ethics.

scholarly journals Duas fotografias em Austerlitz, de W. G. Sebald | Two photographs in Austerlitz, by W. G. Sebald

2021 ◽  
Vol 3 (2) ◽  
pp. 101-116
Author(s):  
Luciano Gatti
Keyword(s):  
Present Article ◽  
Photographic Material ◽  
Starting Point ◽  
Memory Experience ◽  
A Chain ◽  
The Relationship

O presente trabalho discute as relações entre literatura e fotografia em Austerlitz, de W. G. Sebald. Para fazer isso, como ponto de partida, observa-se a suposta oposição entre documento e ficção no emprego de material fotográfico feito por Sebald. Segundo a hipótese deste artigo, a função das fotografias deve ser compreendida a partir do mecanismo literário desenvolvido por Sebald para apresentar a investigação de seu protagonista a respeito de elementos de sua vida passada desconhecida. A pesquisa caracteriza tal procedimento como um “encadeamento de narradores” e, feito isso, debate sobre a função exercida pelas fotografias nas relações entre memória, narração e experiência.Palavras-chave: W. G. Sebald. Fotografia. Memória. Experiência.  AbstractThis article discusses the relationship between literature and photography in Austerlitz, by WG Sebald. In order to do that, as a starting point, we observe the supposed opposition between document and fiction in Sebald's use of photographic material. This study proposes that we may understand the role played by photographs in the book by means of the literary mechanism developed by Sebald to present the search of his protagonist for elements of his unknown past. The present article characterizes this procedure as a “chain of narrators” and, after that, discusses the role played by photographs in the relationships between memory, narration and experience.Keywords: W. G. Sebald. Photography. Memory. Experience. ORCIDhttp://orcid.org/0000-0003-3960-3610

scholarly journals Classification of chain rings

2021 ◽  
Vol 7 (4) ◽  
pp. 5106-5116
Author(s):  
Yousef Alkhamees ◽  
◽  
Sami Alabiad
Keyword(s):  
Artinian Ring ◽  
Chain Ring ◽  
A Chain

<abstract><p>An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.</p></abstract>

scholarly journals Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

2021 ◽  
pp. 137-154
Author(s):  
Hezron Saka Were ◽  
Maurice Oduor Owino ◽  
Moses Ndiritu Gichuki
Keyword(s):  
Maximal Ideal ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))5 = (0); (Z(R))4 ̸= (0) and determine the structures of the unit groups of R for all its characteristics.

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