Series Rings Research Articles

Second-Order Microring Filter with Large Free-Spectral-Range and Wavelength-Tunable-Range over 50 nm

The high-order microring filter has been proposed for a larger free spectral range (FSR) compared with the single microring filter; therefore, it has great potential to be used in wavelength division multiplexing (WDM) systems. In this article, we have designed and fabricated a second-order microring filter made up of two rings connected in series with two Ti thermal heaters deposited above them. The large FSR of 56.8 nm is obtained by decreasing the difference of the radii between the two series rings, achieving similar FSRs to that of higher-order filters but with a simpler and more compact design. The average electrical tuning efficiencies of the two heaters are 0.186 nm/mW and 0.207 nm/mW, and the center wavelength of the filter can be tuned over the entire FSR with an applied electrical power of less than 40 mW.

PRIME SERIAL RINGS WITH KRULL DIMENSION

A Characterization of Prime Serial rings with Krull dimention as rings with a unique maximum ideal. A description of the laltice of deals and laltice submodules of a module over ring with Krull dimension is a direct sum of indecomposable modules.

Rings Whose Certain Modules Are Dual Self-CS-Baer

In this work, we characterize some rings in terms of dual self-CS-Baer modules (briefly, dsCS-Baer modules). We prove that any ring $\Sigma$ is a left and right artinian serial (briefly, as-) ring with $J^2=0$ iff $\Sigma\oplus Z$ is dsCS-Baer for every right $\Sigma$-module $Z$. If $\Sigma$ is a commutative ring, then we prove that $\Sigma$ is an as-ring iff $\Sigma$ is perfect and every $\Sigma$-module is a direct sum of (cyclic) dsCS-Baer $\Sigma$-modules. Also, we show that $\Sigma$ is a right perfect ring iff all countably generated free right $\Sigma$-modules are dsCS-Baer.

On rings admitting nonzero homomorphisms between non-projective modules

In a recent paper, Türkoğlu have studied the rings admitting nonzero homomorphisms between any two non-projective modules (property (T)). His work raises the following question: What are the rings with nonzero maps between non-projective right modules and singular simple modules? As a generalization of the property (T), we consider two families of rings: those rings with nonzero maps from any non-projective module to any singular simple module (property (T1)) and those rings with nonzero maps from any singular simple module to any non-projective module (property (T2)). Complete characterizations of both classes of rings are obtained and it is shown, in particular, that the rings satisfying property (T) are precisely those rings satisfying both properties (T1) and (T2). We show that R satisfies (T) if and only if R has a unique singular simple right R-module, and R is either two-sided Artinian hereditary serial or right completely coretractable. Furthermore, we prove that a commutative ring R has the property (T) if and only if there is a ring decomposition , where A is semisimple Artinian and B satisfies one of the following conditions: (1) B is a semi-Artinian max ring with unique singular simple right R-module. (2) B is an Artinian hereditary serial ring with unique singular simple right R-module.

The opposite of injectivity by proper classes

Proper classes (or exact structures) offer rich research topics due to their important role in category theory. Motivated by the studies on opposite of injective modules, we introduce a new approach to opposed to injectivity in terms of injectively generated proper classes. The smallest possible proper class generated injectively by a single module is the class of all split short exact sequences. We call a module M ι-indigent if the proper class injectively generated by M consists only of split short exact sequences. We are able to show that if R is a ring which is not von Neumann regular, then every right (pure-injective) R-module is either injective or ι-indigent if and only if R is an Artinian serial ring with J 2 (R) = 0 and has a unique non-injective simple right R-module up to isomorphism. Moreover, if R is a ring such that every simple right R-module is pure-injective, then every simple right R-module is ι-indigent or injective if and only if R is either a right V -ring or R = A × B, where A is semisimple, and B is an Artinian serial ring with J 2 (B) = 0. We investigate the class ι(R) which consists of those proper classes such that is injectively generated by a module. We call such a class (right) proper injective profile of a ring R. We prove that if R is an Artinian serial ring with J 2 (R) = 0, then |ι(R)| = 2 n , where n is the number of non-isomorphic non-injective simple right R-modules. In addition, if ι(R) is a chain, then R is a right Noetherian ring over which every right R-module is either projective or i-test, and has a unique singular simple right R-module. Furthermore, in this case, R is either right hereditary or right Kasch. We observe that |ι(R)| ≠ 3 for any ring R which is not von Neumann regular. We construct a bounded complete lattice structure on ι(R) in case ι(R) is a partially ordered set under set inclusion. Moreover, if R is an Artinian serial ring with J 2 (R) = 0, then this lattice structure is Boolean.

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.

Asymptotic Behavior of Kac Polynomials

We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the global behavior of the coefficients of Kac polynomials. All computations were made using SageMath.

On the Restricted Minimum Condition for Rings

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition ( $${\mathrm{r.RMC}}$$ , for short) if R/A is an Artinian right R-module for any essential right ideal A of R. It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with $${\mathrm{r.RMC}}$$ quasi-Frobenius? (ii) Whether a serial ring with $${\mathrm{r.RMC}}$$ must be Noetherian? We carry out a study of rings with $${\mathrm{r.RMC}}$$ and determine when a right extending ring has $${\mathrm{r.RMC}}$$ in terms of rings $${\begin{bmatrix} S&{}\quad M\\ 0&{}\quad R\end{bmatrix}}$$ such that S is right Artinian, $$M_{Q}$$ is semisimple ( $$Q={\mathrm{Q}}(R)$$ ) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with $${\mathrm{r.RMC}}$$ is quasi-Frobenius if and only if $$\hbox {Z}_{r}(R) = \hbox {Z}_{l}(R)$$ if and only if $$\hbox {Z}_{r}(R)$$ is a finitely generated left ideal and $${\mathrm{N}}(R)\cap {\mathrm{Soc}}(R_{R})$$ is a finitely generated right ideal. Right serial rings with $${\mathrm{r.RMC}}$$ are studied and proved that a non-singular serial ring has $${\mathrm{r.RMC}}$$ if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that $$\mathrm{RMC}$$ is not symmetric for a ring.

On Nagata rings with amalgamation properties

Let [Formula: see text] and [Formula: see text] be two commutative rings with unity, let [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. In this paper, we give a characterization for the amalgamated algebra [Formula: see text] to be a Nagata ring, a strong S-domain, and a catenarian. Also, we investigate the conditions that the ring of Hurwitz series over [Formula: see text] has a complete comaximal factorization.

Rings Characterized via Some Classes of Almost-Injective Modules

In this paper, we study rings with the property that every cyclic module is almost-injective (CAI). It is shown that R is an Artinian serial ring with $$J(R)^2=0$$ if and only if R is a right CAI-ring with the finitely generated right socle (or I-finite) if and only if every semisimple right R-module is almost injective, $$R_R$$ is almost injective and has finitely generated right socle. Especially, R is a two-sided CAI-ring if and only if every (right and left) R-module is almost injective. From this, we have the decomposition of a CAI-ring via an SV-ring for which Loewy (R) $$\le $$ 2 and an Artinian serial ring whose squared Jacobson radical vanishes. We also characterize a Noetherian right almost V-ring via the ring for which every semisimple right R-module is almost injective.

On rings with envelopes and covers regarding to C3, D3 and flat modules

In this paper, by taking the class of all [Formula: see text] (or [Formula: see text]) right [Formula: see text]-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes) and a right [Formula: see text]-ring (or a right noetherian [Formula: see text]-ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes). By using isosimple-projective preenvelope, we obtained that if [Formula: see text] is a semiperfect right FGF ring (or left coherent ring), then every isosimple right [Formula: see text]-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.

A Note on z-Ideals and z°-Ideals of the Formal Power Series Rings and Polynomial Rings in an Infinite Set of Indeterminates

Let A be a ring. In this paper we generalize some results introduced by Aliabad and Mohamadian. We give a relation between the z-ideals of A and those of the formal power series rings in an infinite set of indeterminates over A. Consider A[[XΛ]]3 and its subrings A[[XΛ]]1, A[[XΛ]]2, and A[[XΛ]]α, where α is an infinite cardinal number. In fact, a z-ideal of the rings defined above is of the form I + (XΛ)i, where i = 1, 2, 3 or an infinite cardinal number and I is a z-ideal of A. In addition, we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients. As a natural result, we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.

On a new invariant determining the isomorphism classes of $$\Lambda $$-modules with $$\lambda =3$$

For a prime number p and the ring of power series $$\Lambda ={\mathbb {Z}}_p[[T]]$$ , we define a new invariant which determines the isomorphism classes of finitely generated torsion $$\Lambda $$ -modules with $$\lambda =3$$ . We apply this result to the classical Selmer group attached to an elliptic curve over $${\mathbb {Q}}$$ .

Right serial skew Laurent series rings

Let [Formula: see text] be a ring and [Formula: see text] its automorphism. It is proved that the skew Laurent series ring [Formula: see text] is a right serial ring if and only if [Formula: see text] is a right serial right Artinian ring.

Laurent series over

A Heyting field is a nontrivial commutative local ring such that each noninvertible element is zero. It is the most popular constructive substitute for the classical notion of a field. The prototype example is the ring of real numbers. A weak Heyting field, defined to be a Heyting field minus the local requirement, is classically a Heyting field. We show that the ring of Laurent series (or Puiseux series) over any Heyting field is a weak Heyting field, but the ring of Laurent series over is not a Heyting field.

Virtually uniserial modules and rings

We study the class of virtually uniserial modules and rings as a nontrivial generalization of uniserial modules and rings. An R-module M is virtually uniserial if for every finitely generated submodule 0≠K⊆M, K/Rad(K) is virtually simple. Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules and a left virtually uniserial (resp., left virtually serial) ring is a ring which is virtually uniserial (resp., serial) as a left R-module. We give some useful properties of virtually (uni)serial modules and rings. In particular, it is shown that every left virtually uniserial module is uniform and Bézout. Also, we show that if R is a left virtually serial ring, then R/J(R)≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain. As a consequence, we obtain that a ring R is left virtually serial with J(R)=0 if and only if R≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain with J(Di)=0. Also, several classes of rings for which every virtually uniserial module (resp., ring) is uniserial are given. Noetherian left virtually uniserial rings are characterized. Finally, we obtain some structure theorems for (commutative) rings over which every (finitely generated) module is virtually serial.

On a problem concerning the ring of Nash germs and the Borel mapping

We denote by $\mathbb{R}$[[t]] the ring of formal power series with real coefficients. Let $\widehat{\mathcal{C}_{1}}\subset\mathbb{R}[[t]]$ be a subring. We say that $\widehat{\mathcal{C}_{1}}$ has the splitting property if for each $f\in\widehat{\mathcal{C}_{1}}$ and $A \cup B = \mathbb{N}$ such that $A\cap B = \emptyset$, if $f = G+H$ where $G = \displaystyle\sum_{w\in A}a_{w}t^{w}$ and $H = \displaystyle\sum_{w\in B}a_{w}t^{w}$ are formal power series, then $G\in\widehat{\mathcal{C}_{1}}$ and $H\in\widehat{\mathcal{C}_{1}}$. It is well known that the ring of convergent power series $\mathbb{R}${t} satisfies the splitting property. In this paper, we will examine this property for a subring of $\mathbb{R}${t} and for some local rings containing strictly $\mathbb{R}${t}.

On a graph of ideals of a commutative ring

In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I∩J)=ann(I)+ann(J). In this article, the basic properties and possible structures of the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for which G(R) is planar, complete and complete r-partite. We show that, if (R,M) is a local Artinian ring, then G(R) is complete if and only if Soc(R) is simple. Also, it is shown that if R is a ring with G(R) is r-regular, then either G(R) is complete or null graph. Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.

Cancellation and regular derivations

In this paper, we establish a criterion for every complete separated linearly topologized ring to be isomorphic to a ring of power series, i.e. to be a cylinder over another complete topological ring. We use this criterion to establish a general statement of cancellation in the category of local complete but non-necessarily adic topological [Formula: see text]-algebras.

Оцінка властивостей зміцнених нанопокриттям маслоз'ємних поршневих кілець оптико-математичним методом

The paper presents an optical-mathematical evaluation of the physic-mechanical properties of the D100 diesel-generator oil scraper piston ring surface hardened by multi-layer ion-plasma nanocoating TiN/CrN. The application parameters of multilayer nanocoating and the sequence of its layers are shown. The values of wear and microhardness of the hardened and serial rings are given. The increase in wear resistance of the hardened product is shown to be 12 times in weight and 4 times in the degree of increase in the size of its friction surface. A method for evaluating properties using the mathematical operators of the laplacian and divergences in their different variations is proposed. A correlation between the properties of the results of tests for friction and wear with using a mathematical description of structure formation was revealed. The minimal difference of the friction zone and the initial surface of the hardened ring, which does not exceed 16 %, is revealed. The difference between the initial surface and the friction zone in the serial ring is shown, which reaches up to 42 % on average in terms of indicators. A substantial deviation of the values of the friction zone mathematical operators of the serial ring from the TiN/CrN hardened by ion-plasma nanocoating was confirmed, which in some cases reaches up to 42 % on average.