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.

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.

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.

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.

On powers of ideals generated by R-sequences in a Noetherian local ring

1973 ◽  
Vol 74 (3) ◽  
pp. 441-444 ◽  
Author(s):  
A. Caruth
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Primary Decomposition ◽  
Finite Intersection ◽  
Noetherian Local Ring ◽  
Powers Of Ideals ◽  
Primary Ideals ◽  
Unique Maximal Ideal

In a Noetherian commutative ring with identity, every ideal can be expressed (not necessarily uniquely) as a finite intersection of primary ideals (called a primary decomposition). This note is concerned with powers of ideals generated by subsets of an R-sequence in a local ring R (i.e. a Noetherian commutative ring R with identity possessing a unique maximal ideal m) and with a decomposition of such ideals.

scholarly journals On the Zero Divisor Graphs of a Class of Commutative Completely Primary Finite Rings

2016 ◽  
Vol 12 (3) ◽  
pp. 6021-6026
Author(s):  
Maurice Oduor ◽  
Walwenda Shadrack Adero
Keyword(s):  
Maximal Ideal ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

Let R be a Completely Primary Finite Ring with a unique maximal ideal Z(R)), satisfying ((Z(R))n−1 ̸= (0) and (Z(R))n = (0): The structures of the units some classes of such rings have been determined. In this paper, we investigate the structures of the zero divisors of R:

Projective Ideals of Finite Type

1969 ◽  
Vol 21 ◽  
pp. 1057-1061 ◽  
Author(s):  
William W. Smith
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Finite Type ◽  
Local Ring ◽  
Maximal Ideal ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Unique Maximal Ideal

The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used.Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC.Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RPis the quotient ring formed using the complement of P.

scholarly journals Unit groups of cube radical zero commutative completely primary finite rings

2005 ◽  
Vol 2005 (4) ◽  
pp. 579-592
Author(s):  
Chiteng'a John Chikunji
Keyword(s):  
Positive Integer ◽  
Maximal Ideal ◽  
Finite Ring ◽  
Finitely Generated ◽  
Finite Rings ◽  
Group Of Units ◽  
Generating Sets ◽  
Zero Divisors ◽  
Unique Maximal Ideal

A completely primary finite ring is a ringRwith identity1≠0whose subset of all its zero-divisors forms the unique maximal idealJ. LetRbe a commutative completely primary finite ring with the unique maximal idealJsuch thatJ3=(0)andJ2≠(0). ThenR/J≅GF(pr)and the characteristic ofRispk, where1≤k≤3, for some primepand positive integerr. LetRo=GR(pkr,pk)be a Galois subring ofRand let the annihilator ofJbeJ2so thatR=Ro⊕U⊕V, whereUandVare finitely generatedRo-modules. Let nonnegative integerssandtbe numbers of elements in the generating sets forUandV, respectively. Whens=2,t=1, and the characteristic ofRisp; and whent=s(s+1)/2, for any fixeds, the structure of the group of unitsR∗of the ringRand its generators are determined; these depend on the structural matrices(aij)and on the parametersp,k,r, ands.

Non-commutative unique factorization rings with zero-divisors

2020 ◽  
pp. 2150022
Author(s):  
A. Naser ◽  
M. H. Fahmy ◽  
A. M. Hassanein
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Unique Factorization ◽  
Commutative Case ◽  
Zero Divisors

It was shown by Galovich that if [Formula: see text] is a commutative unique factorization ring (UFR) with identity, then [Formula: see text] is a local ring with a nil maximal ideal. In this paper, we generalize Galovich’s results to the non-commutative case.

scholarly journals Finite completely primary rings in which the product of any two zero divisors of a ring is in its coefficient subring

1994 ◽  
Vol 17 (3) ◽  
pp. 463-468
Author(s):  
Yousif Alkhamees
Keyword(s):  
Maximal Ideal ◽  
Galois Ring ◽  
Primary Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Special Case ◽  
Unique Maximal Ideal

According to general terminology, a ringRis completely primary if its set of zero divisorsJforms an ideal. LetRbe a finite completely primary ring. It is easy to establish thatJis the unique maximal ideal ofRandRhas a coefficient subringS(i.e.R/Jisomorphic toS/pS) which is a Galois ring. In this paper we give the construction of finite completely primary rings in which the product of any two zero divisors is inSand determine their enumeration. We also show that finite rings in which the product of any two zero divisors is a power of a fixed prime p are completely primary rings with eitherJ2=0or their coefficient subring isZ2nwithn=2or3. A special case of these rings is the class of finite rings, studied in [2], in which the product of any two zero divisors is zero.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

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