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Let R be a commutative ring and w be the so-called w-operation on R. Then we introduce and study two concepts of w-FP-injective modules and w-IF rings. To do so, we use two main methods, of which o...

The effect of Ce substitution on the structural and magnetic properties of (Nd1-δCeδ)2Fe14B (0

In this paper, we obtain that a commutative ring R is w-coherent if and only if HomR(M,E) is w-flat for any absolutely pure w-module M and any injective (w-)module E, if and only if HomR(M,E) is w-...

In this article, a new semistar operation, called the q-operation, on a commutative ring R is introduced in terms of the ring Q0(R) of finite fractions. It is defined as the map q:Fq(R)→Fq(R) by A↦...

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the $w$-version of this theorem, where $w$ is a hereditary torsion theory for modules over a commutative ring., Comment: 16 pages

An R-module M is called strongly FP-injective if ExtRi(P,M)=0 for any finitely presented R-module P and any i > 0. Denoted by SFI the class of all strongly FP-injective R-modules and by ⊥SFI the le...

In this article, we introduce a new type of projective modules, called the weak w-projective module. By using this type of modules, we give a homological characterization of Krull domains. More pre...

In this paper, we characterize Prüfer [math] -multiplication domains as integral domains which have the property that the existence of a generalized solution of any system of linear equations is equivalent to a weak equality of the determinantal ideals of the coefficient matrix and the augmented matrix of the system. In fact, we obtain a more general result for commutative rings of weak global [math] -dimension (in the sense of Bueso, Van Ostaeyen and Verschoren) at most one, where [math] is a half-centered hereditary torsion theory.

In 2D magnets, interlayer exchange coupling is generally weak due to the van der Waals layered structure but it still plays a vital role in stabilizing the long-range magnetic ordering and determining the magnetic properties. Using complementary neutron diffraction, magnetic, and torque measurements, the complete magnetic phase diagram of CrPS

The carbon doped RMn1.7Cr0.3Si2C (R = Nd, Sm, Dy) compounds were prepared by arc melting and investigated using room temperature (RT) X-ray diffraction (XRD), neutron powder diffraction (NPD) and magnetic measurements. Carbon atoms were found to occupy the interstitial site (0, 0, 0.125) in a superstructure with the space group P4/mmm at room temperature. There exists a disordered substitution between the R atoms and corresponding Si dumbbells on the same layer perpendicular to c direction in the carbides. The Mn/Cr sublattice of the carbides exhibits ferromagnetic characteristics at room temperature despite of a very short intralayer interatomic distance (dMn-Mn = 2.796 A for R = Dy), which is far below the critical distance for a ferromagnetic (dMn-Mn >2.87 A) ordering of Mn sublattice due to the strong hybridization between the interstitial C and the Mn/Cr atoms. Different phase transitions of the carbides have different response to the external magnetic field. These carbides provide the potential for further research on magneto-transport properties and magnetocaloric effect. Additionally, spin reorientation was observed in SmMn1.7Cr0.3Si2C owing to the competence of magnetocrystalline anisotropy between the Mn/Cr sublattice and the Sm sublattice.

Let R be a commutative ring. The Gorenstein super finitely presented dimension of R is defined as $$G\hbox {-}\text{ s.gl. }\dim (R)=\sup \{\text{ Gpd }_{R}{M}\,|\,M$$ is a super finitely presented R-module}. In this paper, we give a series of characterizations of the ring with $$G\text{- }\text{ s.gl }.\dim (R)=0$$ . We show that R is a ring with $$G\text{- }\text{ s.gl }.\dim (R)=0$$ if and only if R is self-weak injective and R is a generalized coherent ring (i.e., R is a GFC-ring), where an R-module M is called weak injective if $$\mathrm{Ext}_{R}^{1}(N,M)=0$$ for every super finitely presented R-module N, and R is called a generalized coherent ring if $$M^{*}$$ is super finitely presented for any super finitely presented R-module M. Further, comparing with FGF-question, we give an example to show that a ring R is not necessarily a QF-ring if every (super) finitely presented R-module can embedded in a free R-module.

In this paper, we concern the Principal Ideal Theorem (PIT) for w-Noetherian rings. Let R be a w-Noetherian ring and a be a nonzero nonunit element of R. If p is a prime ideal of R minimal over (a), then ht p ≦ 1.

It is well known that the equivalence of injectivity and divisibility characterizes the Dedekind domains. In this note, we shed more light on this problem and show that a similar property characterizes Krull domains and π-domains.

Let R be a commutative ring, Q0(R) be the ring of finite fractions over R, and w be the so-called w-operation on R. In this article, we introduce a new type of Prufer v-multiplication ring, called a quasi-Q0-PvMR and defined as a ring R for which every w-linked Q0-overring of R is integrally closed in Q0(R). Our primary motivation for investigating quasi-Q0-PvMRs is to provide w-theoretic analogues to some work of Lucas [16] concerning Q0-Prufer rings. (A ring R is called a Q0-Prufer ring if every Q0-overring of R is integrally closed in Q0(R).)

In this article, we study the hereditary torsion theory defined by the set of associated primes of principle ideals of an integral domain, which is called the g-torsion theory. We first discuss some general properties of g-torsion theories, and after that give some applications of them. For example, we generalize a characterization of reflexive modules over quasi-normal domains to a class of non-Noetherian domains. Among other things, a characterization of coherent domains of weak Gorenstein global dimension at most two is also given in terms of Gorenstein projectivity (or Gorenstein flatness) of injective modules relative to the g-torsion theory.

The quaternary interstitial nitrides Nd2Fe14BNδ was synthesized through the gas-solid reaction of N2 gas and Nd2Fe14B powders. It was found that Nd2Fe14BNδ alloy maintains the tetragonal structure (space group P42/mnm) with an expansion of the unit cell. Structure analyses showed the nitrogen concentration (δ) is about 0.352(48) with the volume-expansion ratio (ΔV/V) of 0.6613% in the unit cell. Refinement of neutron diffraction data indicated that the N has a site occupation preference (8:3) for the 4e (0, 0, 0.0652) site over the 4f (0.1000, 0.1000, 0) site. The room temperature saturation magnetization and the magnetocrystalline anisotropy field of Nd2Fe14BNδ slightly decrease by about 6% and 5% as compared to the parent alloy, respectively. The Curie temperature of Nd2Fe14B compound increases from 312 °C to 331 °C after nitrogenation with such a subtle volume expansion, which provides a validation prediction that we may increase the volume-expansion ratio to improve Tc further. Therefore, Nd2Fe14BNδ, with better temperature stability could be obtained.

In 2D magnets, interlayer exchange coupling is generally weak due to the van der Waals layered structure but it still plays a vital role in stabilizing the long-range magnetic ordering and determining the magnetic properties. Using complementary neutron diffraction, magnetic, and torque measurements, the complete magnetic phase diagram of CrPS4 crystals is determined. CrPS4 shows an antiferromagnetic ground state (A-type) formed by out-of-plane ferromagnetic monolayers with interlayer antiferromagnetic coupling along the c axis below TN = 38 K. Due to small magnetic anisotropy energy and weak interlayer coupling, the low-field metamagnetic transitions in CrPS4, that is, a spin-flop transition at ≈0.7 T and a spin-flip transition from antiferromagnetic to ferromagnetic under a relatively low field of 8 T, can be realized for H∥c. Intriguingly, with an inherent in-plane lattice anisotropy, spin-flop-induced moment realignment in CrPS4 for H∥c is parallel to the quasi-1D chains of CrS6 octahedra. The peculiar metamagnetic transitions and in-plane anisotropy make few-layer CrPS4 flakes a fascinating platform for studying 2D magnetism and for exploring prototype device applications in spintronics and optoelectronics.

In this article, it is proved that a domain R is a Prufer domain if and only if it is coherent, integrally closed and FP-id R (R) ≤ 1.

In this paper, we introduce and study the w-weak globaldimension w-w.gl.dim(R) of a commutative ring R. As an application, itis shown that an integral domain R is a Pru¨fer v-multiplication domainif and only if w-w.gl.dim(R) 6 1. We also show that there is a largeclass of domains in which Hilbert’s syzygy Theorem for the w-weak globaldimension does not hold. Namely, we prove that if Ris an integral domain(but not a field) for which the polynomial ring R[x] is w-coherent, thenw-w.gl.dim(R[x]) = w-w.gl.dim(R). 1. IntroductionRecall that a Pru¨fer domain is an integral domain in which every finitelygenerated ideal is invertible. It is well-known that the concept of Pru¨fer do-mains has played a central role in the development of the classical ideal theory.An important generalization of the Pru¨fer domain notion is that of a Pru¨ferv-multiplication domain (PVMD). This notion comes from multiplicative idealtheory, and various ideal-theoretic properties of it have been considered bymany authors, see for example [1, 2, 4, 7, 9, 10, 13, 14, 15]. From the homolog-ical algebra point of view, Pru¨fer domains are exactly the integral domains ofweak global dimension at most one. The original motivation for this work is toprovide a homological algebra characterization of PVMDs. To do so, we needthe notion of a w-flat module. Recently, modules of this type have receivedattention in several papers in the literature, see for example [3, 11, 21, 22].Throughout, R denotes a commutative ring with an identity element and allmodules are unitary.Now, we review some definitions and notation. Let J be an ideal of R.Following [25], J is called a Glaz-Vasconcelos ideal (a GV-ideal for short) if J isfinitely generated and the natural homomorphism ϕ : R → J

An SM domain is an integral domain which satisfies the as-cending chain condition on w-ideals. Then an SM domain also satisfiesthe descending chain condition on those chains of v-ideals whose intersec-tion is not zero. In this paper, a study is begun to extend these propertiesto commutative rings with zero divisors. A Q 0 -SM ring is defined to be aring which satisfies the ascending chain condition on semiregular w-idealsand satisfies the descending chain condition on those chains of semiregularv-ideals whose intersection is semiregular. In this paper, some propertiesof Q 0 -SM rings are discussed and examples are provided to show the dif-ference between Q 0 -SM rings and SM rings and the difference betweenQ 0 -SM rings and Q 0 -Mori rings. 1. IntroductionIn this paper, we assume that Ris a commutative ring with an identity andSis a multiplicatively closed set of R. Let Z(R) denote the set of zero divisorsof Rand let T(R) be the total quotient ring of R.Let I be an ideal of R. Then I is called a regular ideal if I contains aregular element and I is called a semiregular ideal of Rif I contains a finitelygenerated ideal I

Let R be a ring. A left R-module M (resp., right R-module N) is called weak injective (resp., weak flat) if (resp., ) for every super finitely presented left R-module F. By replacing finitely presented modules by super finitely presented modules, we may generalize many results of a homological nature from coherent rings to arbitrary rings. Some examples are given to show that weak injective (resp., weak flat) modules need not be FP-injective (resp., not flat) in general. In addition, we introduce and study the super finitely presented dimension (denote by l.sp.gldim(R)) of R that are defined in terms of only super finitely presented left R-modules. Some known results are extended.

Let R be a commutative ring, M be an R-module, and w be the so-called w-operation on R. Set S w = { f ∈ R [ X ] | c ( f ) w = R } , where c ( f ) denotes the content of f. Let R { X } = R [ X ] S w and M { X } = M [ X ] S w be the w-Nagata ring of R and the w-Nagata module of M respectively. Then we introduce and study two concepts of w-projective modules and w-invertible modules, which both generalize projective modules. To do so, we use two main methods of which one is to localize at maximal w-ideals of R and the other is to utilize w-Nagata modules over w-Nagata rings. In particular, it is shown that an R-module M is w-projective of finite type if and only if M { X } is finitely generated projective over R { X } ; M is w-invertible if and only if M { X } is invertible over R { X } . As applications, it is shown that R is semisimple if and only if every R-module is w-projective and that, in a Q 0 -PVMR, every finite type semi-regular module is w-projective.

In this paper, we characterize w-Noetherian modules in terms of polynomial modules and w-Nagata modules. Then it is shown that for a finite type w-module M , every w-epimorphism of M onto itself is an isomorphism. We also define and study the concepts of w-Artinian modules and w-simple modules. By using these concepts, it is shown that for a w-Artinian module M , every w-monomorphism of M onto itself is an isomorphism and that for a w-simple module M , EndRM is a division ring.

In this paper, we prove that the Gorenstein analogue of the well-known Auslander's theorem on the global dimension holds true. Namely, we prove that the Gorenstein global dimension of a commutative ring R is equal to the supremum of the set of Gorenstein projective dimensions of all cyclic R-modules.

Let Rbe a commutative ring with identity. An R-module Mis said to be w-projective if Ext 1R (M,N) is GV-torsion for any torsion-freew-module N. In this paper, we define a ring R to be w-semi-hereditaryif every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules andstudy some basic properties of w-injective modules. Using these concepts,we show that Ris w-semi-hereditary if and only if the total quotient ringT(R) of R is a von Neumann regular ring and R m is a valuation domainfor any maximal w-ideal mof R. It is also shown that a connected ring Ris w-semi-hereditary if and only if Ris a Pru¨fer v-multiplication domain. 1. IntroductionThroughout, R denotes a commutative ring with identity 1 and E(M) de-notes the injective hull (or envelope) of an R-module M. And let us regardthat the v-, t- and w-operation are well-known star-operations on domains. Forunexplained terminologies and notations, we refer to [3, 14, 15].Pru¨fer v-multiplication domains (PVMD for short) have received a good dealof attention in much literature. A domain R is called a PVMD if every nonzerofinitely generated ideal I is t-invertible, that is, there is a fractional ideal B ofR such that (IB)

Mn-based Heusler alloys have attracted significant research attention as half-metallic materials because of their giant magnetocrystalline anisotropy and magnetocaloric properties. We investigate the crystal structure and magnetic properties of polycrystalline, [101]-oriented, and [100]-oriented Mn 2 − δ Sn prepared separately by arc melting, the Bridgeman method, and the flux method. All of these compounds crystallize in a Ni2In-type structure. In the Mn 2 − δ Sn lattice, Mn atoms occupy all of the 2a and a fraction of the 2d sites. Site disorder exists between Mn and Sn atoms in the 2c sites. In addition, these compounds undergo a re-entrant spin-glass-like transition at low temperatures, which is caused by frustration and randomness within the spin system. The magnetic properties of these systems depend on the crystal directions, which means that the magnetic interactions differ significantly along different directions. Furthermore, these materials exhibit a giant magnetocaloric effect near the Curie temperature. The largest value of maximum of magnetic entropy change ( − Δ S M ) occurs perpendicular to the [100] direction. Specifically, at 252 K, maximum − Δ S M is 2.91 and 3.64 J · kg − 1 K − 1 for a magnetic field of 5 and 7 T, respectively. The working temperature span over 80 K and the relative cooling power reaches 302 J/kg for a magnetic field of 7 T, which makes the Mn 2 − δ Sn compound a promising candidate for a magnetic refrigerator.

In this paper, we introduce the class of quasi-strongly Gorenstein projective modules which is a particular subclass of the class of finitely generated Gorenstein projective modules. We also introduce and characterize quasi-strongly Gorenstein semihereditary rings. We call a quasi-strongly Gorenstein semihereditary domain a quasi-SG-Prüfer domain. A Noetherian quasi-SG-Prüfer domain is called a quasi-strongly Gorenstein Dedekind domain. Let [Formula: see text] be a field and [Formula: see text] be an indeterminate over [Formula: see text]. We prove that every ideal of the ring [Formula: see text] is strongly Gorenstein projective. We also show that every ideal of the ring [Formula: see text] (respectively, [Formula: see text]) is strongly Gorenstein projective. These domains are examples of quasi-strongly Gorenstein Dedekind domains.

Let [Formula: see text] be a commutative ring. In this paper, a class of almost projective modules is introduced. An [Formula: see text]-module [Formula: see text] is said to be almost projective if [Formula: see text] for any [Formula: see text]-module [Formula: see text], where [Formula: see text] is a maximal ideal of [Formula: see text]. It is shown that an [Formula: see text]-module [Formula: see text] satisfying that [Formula: see text] is free over [Formula: see text] for any maximal ideal [Formula: see text] of [Formula: see text] is exactly almost projective. As applications, we characterize almost Dedekind domains in terms of almost projectivity and certain divisibility, respectively.

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].