Ideals In Rings Research Articles

OUR EXPERIENCE OF MITRAL VALVE REPAIR IN A PREDOMINANT RHEUMATIC POPULATION USING A SEMIFLEXIBLE ANNULOPLASTY RING (CE PHYSIO 2 ANNULOPLASTY RING).

Background: Annular dilation is the most typical nding in Mitral regurgitation and annuloplasty with a prosthetic ring is common to various techniques described in literature Mitral valve repair in rheumatic heart disease possesses various technical difculties with no general consensus on the ideal annuloplasty ring. This study describes our experience and results with a semirigid annuloplasty ring (CE physio II ring) in a predominant rheumatic population. This is a retrospectiv Methods: e study using data including 23 patients who underwent mitral valve repair at Vardhman Mahavir Medical College and Safdarjung hospital, New Delhi, between July 2018 to April 2022. Patients were followed up to a mean of 19.3 10.6 months with a minimum follow-up period of 6 months. The mean age was Results: 30.5 15.3 years. The majority of patients, 78.3% (18) had rheumatic heart disease. 82.6% (19) of patients belonged to NYHA class I/II. The results in our study with CE physio II ring are excellent with 91.3% patients free from signicant MR (grade 2). 72% of the patients had improved or preserved Left ventricular function with a decrease in mean LVEF by 2.4% which was statistically insignicant(P>0.05). There was 0 operative or early mortality in our study. The semiexible CE physio II ring provides Conclusions: excellent results in both short and midterm outcomes in terms of freedom from recurrence of signicant MR with preservation of left ventricular function.

A characterization of Krull domains in terms of their factor rings

ABSTRACT Let D be an integral domain, w be the so-called w-operation on D, and be the set of nonzero finitely generated ideals J of D with . Let I be a w-ideal of D, D/I be the factor ring of D modulo I, be the injective envelope of D/I, and . Recently Zhou, Kim, and Hu showed that is a commutative ring with identity containing D/I. The Krull domains are a w-analog of the Dedekind domains in the sense that D is a Krull domain if and only if every nonzero ideal of D is w-invertible. It is well known that D is a Dedekind domain if and only if D/aD is a zero-dimensional semilocal principal ideal ring (PIR) for all nonzero nonunits , if and only if D/I is a zero-dimensional semilocal PIR for all nonzero ideals I of D. In this paper, we prove that D is a Krull domain if and only if is a zero-dimensional semilocal PIR for any nonzero nonunit a of D, if and only if is a zero-dimensional semilocal PIR for any w-ideal I of D. We also study when is Noetherian, Artinian, or a (zero-dimensional) semilocal Bézout ring for all nonzero principal ideals or w-ideals I of D.

On Farrell–Tate cohomology of GL3 over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on mod ℓ Farrell–Tate cohomology for arithmetic groups. We deal with ℓ-rank 1 cases different from PSL2. The conjugacy classification of cyclic subgroups of order ℓ is reduced to the classification of modules of Cℓ-group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell–Tate cohomology computations, we discuss in detail the homological torsion in PGL3 over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

On modules in which every finitely generated submodule is a kernel of an endomorphism

We study an R-module M in which every finitely generated submodule of M is a kernel of an endomorphism of M. Such modules are called Co-epi-finite-retractable (CEFR). We also consider CEFR condition on the co-local modules, submodules and factors of a CEFR module and direct sum of CEFR modules. Among other results, we prove that the injective hull of a simple module over a commutative Noetherian ring is uniserial if and only if it is CEFR.We investigate modules over a principal ideal ring, and show that all finitely generated torsion modules over a principal ideal domain are CEFR. Also, we show that every module over a commutative Köthe ring is CEFR. We also observe that a ring R is left pseudo morphic if and only if it is CEFR as a left R-module and we obtain some new properties of left pseudo morphic rings.

RD-flatness and RD-injectivity of simple modules

We say that a ring R is a right RDV-ring if each simple right R -module is RD -injective. In this paper, we study the notion of RDV -rings which is a non-trivial generalization of V-rings and Köthe rings. For instance, commutative RD -rings, serial rings and right duo right uniserial rings are RDV -rings. Several characterizations of right RDV -rings are given. Also, it is shown that over a semilocal ring R with Jacobson radical J , each simple right R -module is RD -flat if and only if R is a left RDV -ring, if and only if R ( R / J ) is RD -injective, if and only if ( R / J ) R is RD -flat. As a consequence, we show that a local ring R is a principal ideal ring if and only if R satisfies the ascending chain condition on principal left ideals and R ( R / J ) is RD -injective. In the case of R being either a local left perfect ring or a normal left perfect ring, we have obtained results which state that to check whether every left R -module is RD -injective (or, R is Köthe), it suffices to test only the RD -injectivity of the simple left R -modules. Finally, we give some characterizations of quasi-Frobenius rings by using these concepts.

Computing free non-commutative Gröbner bases over [formula omitted] with Singular:Letterplace

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z〈X〉. We present all the needed proofs in details, add a part on the direct treatment of the ring Z/mZ as well as new examples and applications to e.g. Iwahori-Hecke algebras. The extension of Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behaviour of Gröbner bases between free algebras and algebras, which are close to commutative. Even the process of the formation of critical pairs has to be reengineered, together with implementing the criteria for their quick discarding. We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Gröbner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation.

L-LIPs of codes over finite chain rings

In this paper, we first generalize the l-linear intersection pair (l-LIP) of codes over finite fields to the linear l-intersection pair of codes over finite chain rings. Then three criteria of l-linear intersection pair of codes over finite chain rings are obtained. By means of the generator set of cyclic codes, we obtain the generator set of the intersection of two cyclic codes which helps us to get a characterization of l-cyclic intersection pair of codes. As an application, l-cyclic intersection pair of codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting codes. Finally, under suitable conditions, we also obtain a characterization for a l-cyclic intersection pair of codes over principal ideal rings Zk.

Tuning terahertz transitions in cyclo[n]carbon rings

We develop an analytic model for an ideal polyyne ring which describes the induced terahertz (THz) gap in the molecular spectrum due to the Stark effect. This simple model can also be used to describe an odd-dimered cyclocarbon which has undergone a spontaneous symmetry-breaking event (due to the Jahn-Teller effect) as an effective dipole across an ideal ring. We show that both the size of the gap and the strength of optical transitions across it can be modulated by varying the external electric field strength. A THz emission scheme based on optical excitation is proposed, thus paving the way for tunable THz emitters based on $\mathrm{cyclo}[n]\mathrm{carbon}$.

Diagnostic tests for discrete functions defined on rings

Abstract The paper is concerned with sources of faults associated with commutative principal ideal rings. Tables of faults of such sources are known to correspond to Cayley multiplication tables in rings, whose elements are replaced by the values of a Boolean function of these elements. For such rings, the concepts of a diagnostic test and the Shannon function for the length of a diagnostic test are introduced in a natural way. It is shown that if A is a principal ideal ring with only one prime ideal p ≠ A, and if pn = 0 for some n ∈ ℕ, then, for this ring, the Shannon length function of a diagnostic test has the form L diagn(A, n) = Θ(n). We also define an easily testable functions, i.e., a function with respect to which the order of growth of the length of a diagnostic test with respect to this function is equal to the logarithm of the number of pairwise distinct columns of the table of faults. A link between easily testable functions and column separation of tables of faults for two concrete sources of faults is established.

Ideal containment in commutative rings

Let [Formula: see text] be a commutative ring with identity. An ideal [Formula: see text] of [Formula: see text] is said to be a big ideal (respectively, an upper big ideal) if whenever [Formula: see text] (respectively, [Formula: see text]), [Formula: see text] (respectively, [Formula: see text]) for every [Formula: see text]; and [Formula: see text] itself is a big ideal ring provided that every ideal of [Formula: see text] is a big ideal. In this paper, we study the notions of big ideals, upper big ideals and big ideal rings in different contexts of commutative rings such us integrally closed domains, pullbacks and trivial ring extensions. We show that the notions of big and upper big ideals are completely different. The notion of big ideal is correlated to the notion of basic ideal and the notion of upper big ideal is correlated to the notion of [Formula: see text]-ideals. We give a new characterization of Prüfer domains via big ideal domains and we characterize some particular cases of pullback rings that are big ideal domains. Also, we give some classes of big and upper big ideals in rings with zero-divisors via trivial ring extensions.

Ideal nodal rings of one-dimensional photonic crystals in the visible region

Three-dimensional (3D) artificial metacrystals host rich topological phases, such as Weyl points, nodal rings, and 3D photonic topological insulators. These topological states enable a wide range of applications, including 3D robust waveguides, one-way fiber, and negative refraction of the surface wave. However, these carefully designed metacrystals are usually very complex, hindering their extension to nanoscale photonic systems. Here, we theoretically proposed and experimentally realized an ideal nodal ring in the visible region using a simple 1D photonic crystal. The π-Berry phase around the ring is manifested by a 2π reflection phase’s winding and the resultant drumhead surface states. By breaking the inversion symmetry, the nodal ring can be gapped and the π-Berry phase would diffuse into a toroidal-shaped Berry flux, resulting in photonic ridge states (the 3D extension of quantum valley Hall states). Our results provide a simple and feasible platform for exploring 3D topological physics and its potential applications in nanophotonics.

Rees algebra and special fiber ring of binomial edge ideals of closed graphs

In this paper, I compute the regularity of the Rees algebra of binomial edge ideals of closed graphs. I obtain a lower bound for the regularity of the Rees algebra of binomial edge ideals. I also present some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. I obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.

Experimental study on the joint bearing behavior of segmented tower structures subjected to normal and bending shear loads

AbstractThis article deals with the analysis and evaluation of the structural behavior of segmented tower constructions in large‐scale experimental investigations. For this purpose, a tower model with dry horizontal joints on a scale of approximately 1:10 is constructed and loaded. The objective of these large‐scale investigations is to determine the load‐bearing behavior of concrete segment towers subjected to normal force (external prestressing), bending, shear force and torsion and to derive more efficient and realistic design models. The transfer of shear stresses between individual segments is ensured by frictional resistance in the horizontal joints due to prestressing. The current design models are based on plane flange surfaces at the top and bottom of the segments as well as their ideal circular ring shape. This assumes a constant normal stress distribution for the compression connection. Within the large‐scale experimental tests, findings for uneven force distribution in the horizontal joints due to prestressing and shear bending were obtained, that have a significant impact on the design models. However, for the evaluation of the results and the projection onto the real construction components, the scale effects must be urgently taken into account.

Ideal ring extensions and trusses

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of extensions yields an isomorphism of corresponding trusses. Furthermore, equivalence classes of ideal extensions of rings by integers are in one-to-one correspondence with associated trusses up to isomorphism given by a translation. Conversely, to any truss T and an element of this truss one can associate a ring and its extension by integers in which T is embedded as a truss. Consequently any truss can be understood as arising from an ideal extension by integers. The key role is played by interpretation of ideal extensions by integers as extensions defined by double homothetisms of Redei (L. Redei (1952) [24]) or by self-permutable bimultiplications of Mac Lane (S. Mac Lane (1958) [18]), that is, as integral homothetic extensions. The correspondence between homothetic ring extensions and trusses is used to classify fully up to isomorphism trusses arising from rings with zero multiplication and rings with trivial annihilators.

Skew Polynomial Rings: the Schreier Technique

Schreier bases are introduced and used to show that skew polynomial rings are free ideal rings, i.e., rings whose one-sided ideals are free of unique rank, as well as to compute a rank of one-sided ideals together with a description of corresponding bases. The latter fact, a so-called Schreier-Lewin formula (Lewin Trans. Am. Math. Soc.145, 455–465 1969), is a basic tool determining a module type of perfect localizations which reveal a close connection between classical Leavitt algebras, skew polynomial rings, and free associative algebras.

Further results on higher weights of codes over finite rings

The descriptions of generalized Hamming weights with respect to rank are given for codes over chain rings. Based on the descriptions of generalized Hamming weights with respect to rank, the double chain condition, in particular, the chain condition, is introduced and some judging criteria for the double chain condition are presented. As an application of the chain condition, we determine generalized Hamming weights with respect to rank of the tensor product of certain codes satisfying the chain condition. By using generalized Hamming weights with respect to rank, we generalize some results obtained in recent references. We introduce relative generalized Hamming weights to codes over chain rings and principal ideal rings, and present equivalent descriptions and bounds on them. We also generalize maximum distance codes with respect to rank to relative maximum distance codes with respect to rank and give a series of judging criteria for relative maximum distance codes.

Counting non-isomorphic generalized Hamilton quaternions

In this paper we study the isomorphisms of generalized Hamilton quaternions $\Big(\frac{a,b}{R}\Big)$ where $R$ is a finite unital commutative ring of odd characteristic and $a,b \in R$. We obtain the number of non-isomorphic classes of generalized Hamilton quaternions in the case where $R$ is a principal ideal ring. This extends the case $R=\mathbb{Z}/n\mathbb{Z}$
 where $n$ is an odd integer.

On $ S $-principal right ideal rings

<abstract><p>Let $ S $ be a multiplicative subset of a ring $ R $. A right ideal $ A $ of $ R $ is referred to as <italic>$ S $-principal</italic> if there exist an element $ s \in S $ and a principal right ideal $ aR $ of $ R $ such that $ As \subseteq aR \subseteq A $. A ring is referred to as an $ S $-<italic>principal right ideal ring</italic> ($ S $-PRIR) if every right ideal is $ S $-principal. This paper examines $ S $-PRIRs, which extend the notion of principal right ideal rings. Various examples, including several extensions of $ S $-PRIRs are investigated, and some practical results are proven. A noncommutative $ S $-PRIR that is not a principal right ideal ring is found, and the $ S $-variants of the Eakin-Nagata-Eisenbud theorem and Cohen's theorem for $ S $-PRIRs are proven.</p></abstract>

Depth and Stanley depth of the quotient rings of edge ideals of some lobster trees and unicyclic graphs

We compute depth and Stanley depth of the quotient rings of the edge ideals associated with different classes of graphs. These classes include some lobster trees and unicyclic graphs. We show that the values of depth and Stanley depth are equal for the classes we considered.

Depth and Stanley depth of the edge ideals of multi triangular snake and multi triangular ouroboros snake graphs

<abstract><p>In this paper, we study depth and Stanley depth of the quotient rings of the edge ideals associated to triangular and multi triangular snake and triangular and multi triangular ouroboros snake graphs. In some cases, we find exact values, otherwise, we find tight bounds. We also find lower bounds for the edge ideals of triangular and multi triangular snake and ouroboros snake graphs and prove a conjecture of Herzog for all edge ideals we considered.</p></abstract>