The non-orthogonal Cayley–Dickson construction and the octonionic structure of the E8-lattice

2017 ◽  
Vol 16 (12) ◽  
pp. 1750230
Author(s):  
Holger P. Petersson
Keyword(s):  
Commutative Ring ◽  
Linear Form ◽  
Sufficient Conditions ◽  
Algebraic Properties ◽  
Necessary And Sufficient

Using a conic [Formula: see text] algebra [Formula: see text] over an arbitrary commutative ring, a scalar [Formula: see text] and a linear form [Formula: see text] on [Formula: see text] as input, the non-orthogonal Cayley–Dickson construction produces a conic algebra [Formula: see text] and collapses to the standard (orthogonal) Cayley–Dickson construction for [Formula: see text]. Conditions on [Formula: see text] that are necessary and sufficient for [Formula: see text] to satisfy various algebraic properties (like associativity or alternativity) are derived. Sufficient conditions guaranteeing non-singularity of [Formula: see text] even if [Formula: see text] is singular are also given. As an application, we show how the algebras of Hurwitz quaternions and of Dickson or Coxeter octonions over the rational integers can be obtained from the non-orthogonal Cayley–Dickson construction.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

On the Prime Ideals in a Commutative Ring

2000 ◽  
Vol 43 (3) ◽  
pp. 312-319 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Prime Ideals ◽  
Preceding Result ◽  
Finite Commutative Ring ◽  
The Axiom Of Choice ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

Some results on the complement of the annihilating ideal graph of a commutative ring

2015 ◽  
Vol 14 (07) ◽  
pp. 1550099 ◽  
Author(s):  
S. Visweswaran ◽  
Hiren D. Patel
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

scholarly journals COMMUTATIVITY OF SKEW SYMMETRIC ELEMENTS IN GROUP RINGS

2007 ◽  
Vol 50 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Osnel Broche Cristo ◽  
César Polcino Milies
Keyword(s):  
Commutative Ring ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

AbstractLet $R$ be a commutative ring with unity and let $G$ be a group. The group ring $RG$ has a natural involution that maps each element $g\in G$ to its inverse. We denote by $RG^-$ the set of skew symmetric elements under this involution. We study necessary and sufficient conditions for $RG^-$ to be commutative.

scholarly journals More on the annihilator-ideal graph of a commutative ring

2019 ◽  
Vol 18 (08) ◽  
pp. 1950160
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
Star Graph ◽  
Annihilator Ideal ◽  
Vertex Set ◽  
Necessary And Sufficient

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph [Formula: see text] (a well known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with [Formula: see text]. All rings whose [Formula: see text] and [Formula: see text] are characterized. Among other results, we obtain necessary and sufficient conditions under which [Formula: see text] is a star graph.

scholarly journals Pairs of rings with a bijective correspondence between the prime spectra

1993 ◽  
Vol 55 (2) ◽  
pp. 238-245
Author(s):  
E. Jespers ◽  
P. Wauters
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Free Abelian Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Prime Spectrum ◽  
Bijective Correspondence ◽  
Natural Mapping ◽  
Necessary And Sufficient

AbstractLet A be a subring of a commutative ring B. If the natural mapping from the prime spectrum of B to the prime spectrum of A is injective (respectively bijective) then the pair (A, B) is said to have the injective (respectively bijective) Spec-map. We give necessary and sufficient conditions for a pair of rings A and B graded by a free abelian group to have the injective (respectively bijective) Spec-map. For this we first deal with the polynomial case. Let l be a field and k a subfield. Then the pair of polynomial rings (k[X], l[X]) has the injective Spec-map if and only if l is a purely inseparable extension of k.

COMMUTATIVE TALL RINGS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350129
Author(s):  
TOMÁŠ PENK ◽  
JAN ŽEMLIČKA
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Theoretical Criterion ◽  
Necessary And Sufficient

A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.

scholarly journals Extending abelian groups to rings

2007 ◽  
Vol 82 (3) ◽  
pp. 297-314 ◽  
Author(s):  
Lynn M. Batten ◽  
Robert S. Coulter ◽  
Marie Henderson
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Equivalence Relation ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Necessary And Sufficient Conditions ◽  
Odd Order ◽  
Weak Isomorphism ◽  
Necessary And Sufficient

AbstractFor any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

scholarly journals Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jingyu Yang ◽  
Liu Liu ◽  
Yufeng Lu
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Rank Product ◽  
Algebraic Properties ◽  
Necessary And Sufficient ◽  
Product Problem ◽  
Quasihomogeneous Symbols

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.

scholarly journals Some Extensions of Generalized Morphic Rings and EM-rings

2018 ◽  
Vol 26 (1) ◽  
pp. 111-123
Author(s):  
Manal Ghanem ◽  
Emad Abu Osba
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in general true. We put necessary and sufficient conditions to ensure the converse using idealization and polynomial rings

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