Behavior of countably generated pure-projective modules

We first prove that every countably presented module is a pure epimorphic image of a countably generated pure-projective module, and by using this we show that if every countably generated pure-projective module is pure-injective then every module is pure-injective, while if in any countably generated pure-projective module every countably generated pure-projective pure submodule is a direct summand then every module is pure-projective

Some characterizations of regular modules

Let M be a left modula over a ring R. M is called a Zelmanowitz-regular module if for each x Є M there exists a homomorphism f : M → R such that f(x)x = x . Let Q be a left R-module and h : Q → M a homomorphism . We call h locally split if for each x Є M there exists a homomorphism g: M →Q such that h(g(x)) = x . M is called locally projective if every epimorphism onto M is locally split . We prove that the following conditions are equivalent: (1) M is Zelmanowitz-regular. (2) every homomorphism into M is locally split. (3) M is locally projective and every cyclic submodule of M is a direct summand of M

Further results on the inverse along an element in semigroups and rings

In this paper, we introduce a new notion in a semigroup $S$ as an extension of Mary's inverse. Let $a,d\in S$. An element $a$ is called left (resp. right) invertible along $d$ if there exists $b\in S$ such that $bad=d$ (resp. $dab=b$) and $b\leq_\mathcal{L}d$ (resp. $b\leq_\mathcal{R}d$). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) $\pi$-regularity and left (right) $*$-regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed.The authors are highly grateful to the referee for valuable comments which led to improvements of this paper. In particular, Corollaries 2.5, 2.6 and 3.6, Remarks 2.13 and 3.10 and the final remark (ii) were suggested to the authors by the referee. The first author is grateful to China Scholarship Council for giving him a purse for his further study in University of Minho, Portugal. Jianlong Chen and Huihui Zhu are financed by the National Natural Science Foundation of China (No. 11201063 and No. 11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020), the Natural Science Foundation of Jiangsu Province (No. BK20141327), the Foundation of Graduate Innovation Program of Jiangsu Province(No. CXLX13-072), the Scientific Research Foundation of Graduate School of Southeast University and the Fundamental Research Funds for the Central Universities (No. 22420135011). Pedro Patr´ıcio is financed by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, through the Project PEst-OE/MAT/UI0013/2014

Polynomial identities and noncommutative versal torsors

To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle t, we attach two "universal algebras" A(H,t) and U(H,t). The algebra A(H,t) is obtained by twisting the multiplication of H with the most general two-cocycle u formally cohomologous to t. The cocycle u takes values in the field of rational functions on H. By construction, A(H,t) is a cleft H-Galois extension of a "big" commutative algebra B(H,t). Any "form" of H[t] can be obtained from A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] is simple, then A(H,t) is an Azumaya algebra with center B(H,t). The algebra U(H,t) is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H[t] are satisfied. We construct an embedding of U(H,t) into A(H,t); this embedding maps the center Z(H,t) of U(H,t) into B(H,t) when the algebra H[t] is simple. In this case, under an additional assumption, A(H,t) is isomorphic to B(H,t) \otimes_{Z(H,t)} U(H,t), thus turning A(H,t) into a central localization of U(H,t). We work out these constructions in full detail for the four-dimensional Sweedler algebra.Comment: 39 page

N-(2,3-Dimethyl­phen­yl)benzamide

The conformation of the N—H bond in the structure of the title compound, C15H15NO, is anti to the ortho and meta-methyl substituents in the aniline benzene ring, in contrast to the syn conformation observed with respect to the ortho and meta-chloro substituents in N-(2,3-dichloro­phen­yl)benzamide. Furthermore, the conformations of N—H and C=O bonds in the amide group are anti to each other, similar to those observed in other benzanilides. The dihedral angle between the benzoyl and aniline rings is 84.1 (2)°. The amide group is twisted by 23.0 (3)° out of the plane of the benzoyl ring. The structure exhibits positional disorder over the aniline ring, with site occupancies of 0.80 (1) and 0.20 (1) for the major and minor components, respectively. In the crystal, mol­ecules are connected through N—H⋯O hydrogen bonds into chains running along the b axis. An intra­molecular C—H⋯O close contact occurs

Extension groups between atoms and objects in locally noetherian Grothendieck category

We define the extension group between an atom and an object in a locally noetherian Grothendieck category as a module over a skew field. We show that the dimension of the i-th extension group between an atom and an object coincides with the i-th Bass number of the object with respect to the atom. As an application, we give a bijection between the E-stable subcategories closed under arbitrary direct sums and direct summands and the subsets of the atom spectrum and show that such subcategories are also closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms. We show some relationships to the theory of prime ideals in the case of noetherian algebras.Comment: 18 page

Controlling factors of large-scale harmful algal blooms with Karenia selliformis after record-breaking marine heatwaves

Unprecedented, large-scale harmful algal blooms (HABs) dominated by Karenia selliformis occurred off the southeastern coast of Hokkaido, Japan, from late September to early November 2021, about a month after intense and extensive marine heatwaves (MHWs) had subsided. The aims of the present study were to understand the mechanism of development, maintenance, and decay of the HABs as well as to investigate the effect of the MHWs on the HABs. We developed a one-dimensional, lower trophic-level ecosystem model (NEMURO+) to simulate the HABs. The model successfully simulated the 2021 HABs and indicated that their development, maintenance, and decay were controlled primarily by changes of water temperature. Nitrate supply from subsurface layers by seasonal vertical diffusion in autumn also helped to maintain the HABs. Vertical diffusion following MHWs in 2021 contributed to the long duration of the preferred temperature for K. selliformis and the occurrence of pre-bloom of K. selliformis, resulting in preconditioning and accelerating the HABs. However, simulations for normal years (i.e., the climatological mean during 2003–2018) showed that HABs could have occurred, even in the absence of MHWs. The simulations indicated that massive blooms of other phytoplankton species (e.g., diatoms) would not have occurred in 2021, even in the absence of a K. selliformis bloom. The implication was that the HABs in 2021 were the species-specific responses of K. selliformis. The proposed mechanism of the HABs was peculiar to our study area and differed from that previously reported for other K. selliformis blooms. Specifically, the preferred temperature for the HABs of K. selliformis was clearly lower than the previously reported preferred temperature of K. selliformis; thus, the physiological characteristics of the K. selliformis that bloomed in our study area differed from those of other K. selliformis strains. These discoveries provide the first evidence to explain how MHWs affect HABs, and to understand how inter-regional dissimilarities of K. selliformis can lead to large-scale, devastating outbreaks under different oceanographic conditions

One-pot synthesis and AFM imaging of a triangular aramide macrocycle

Macrocyclizations in exceptionally good yields were observed during the self-condensation of N-benzylated phenyl p-aminobenzoates in the presence of LiHMDS to yield three-membered cyclic aramides that adopt a triangular shape. An ortho-alkyloxy side chain on the N-benzyl protecting group is necessary for the macrocyclization to occur. Linear polymers are formed exclusively in the absence of this Li-chelating group. A model that explains the lack of formation of other cyclic congeners and the demand for an N-(o-alkoxybenzyl) protecting group is provided on the basis of DFT calculations. High-resolution AFM imaging of the prepared molecular triangles on a calcite(10.4) surface shows individual molecules arranged in groups of four due to strong surface templating effects and hydrogen bonding between the molecular triangles