Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and on...Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.展开更多
The solubilities of trimethylolethane in butanol,methyl acetate,ethyl acetate as well as in mixed solvents of(methanol+ethyl acetate) and(ethanol+ethyl acetate) were measured with the gravimetric method in the tempera...The solubilities of trimethylolethane in butanol,methyl acetate,ethyl acetate as well as in mixed solvents of(methanol+ethyl acetate) and(ethanol+ethyl acetate) were measured with the gravimetric method in the temperature range from 283.15 K to 318.15 K under atmosphere pressure.The experiment results showed that the solubility of trimethylolethane increased with the temperature,or along with the concentration of methanol or ethanol in the solvents of(methanol+ethyl acetate) and(ethanol + ethyl acetate).In addition,the experiment values were correlated by the van't Hoff equation,Modi fied Apelblat Equation,λh Equation,CNIBS/R-K equation and Jouyban–Acree Model.The Modi fied Apelblat Equation provided the best fitting results of the solubility data of TME in the pure solvents while the CNIBS/R-K model showed the best estimation of the solubility in the binary solvent mixtures.Furthermore,the density functional theory(DFT) calculations showed that solubility in different solvents related to the strength of the interaction between the trimethylolethane and the solvent molecules.Finally,the standard molar enthalpy and molar entropy of trimethylolethane during the dissolving process was also calculated by Modi fied Apelblat equation in this work.展开更多
文摘Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.
基金Supported by the NSFC(21576206)the Program for Changjiang Scholars and Innovative Research Team in University(IRT_15R46)
文摘The solubilities of trimethylolethane in butanol,methyl acetate,ethyl acetate as well as in mixed solvents of(methanol+ethyl acetate) and(ethanol+ethyl acetate) were measured with the gravimetric method in the temperature range from 283.15 K to 318.15 K under atmosphere pressure.The experiment results showed that the solubility of trimethylolethane increased with the temperature,or along with the concentration of methanol or ethanol in the solvents of(methanol+ethyl acetate) and(ethanol + ethyl acetate).In addition,the experiment values were correlated by the van't Hoff equation,Modi fied Apelblat Equation,λh Equation,CNIBS/R-K equation and Jouyban–Acree Model.The Modi fied Apelblat Equation provided the best fitting results of the solubility data of TME in the pure solvents while the CNIBS/R-K model showed the best estimation of the solubility in the binary solvent mixtures.Furthermore,the density functional theory(DFT) calculations showed that solubility in different solvents related to the strength of the interaction between the trimethylolethane and the solvent molecules.Finally,the standard molar enthalpy and molar entropy of trimethylolethane during the dissolving process was also calculated by Modi fied Apelblat equation in this work.
