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Drought is a natural hazard disaster that can deeply affect environments, economies, and societies around the world. Therefore, accurate monitoring of patterns in drought is important. Precipitation is the key variable to define the drought index. However, the spare and uneven distribution of rain gauges limit the access of long-term and reliable in situ observations. Remote sensing techniques enrich the precipitation data at different temporal–spatial resolutions. In this study, the climate prediction center morphing (CMORPH) technique (CMORPH-CRT), the tropical rainfall measuring mission (TRMM) multi-satellite precipitation analysis (TRMM 3B42V7), and the integrated multi-satellite retrievals for global precipitation measurement (IMERG V05) were evaluated and compared with in situ observations for the drought monitoring in the Xiang River Basin, a humid region in China. A widely-used drought index, the standardized precipitation index (SPI), was chosen to evaluate the drought monitoring utility. The atmospheric water deficit (AWD) was used for comparison of the drought estimation with SPI. The results were as follows: (1) IMERG V05 precipitation products showed the highest accuracy against grid-based precipitation, followed by CMORPH-CRT, which performed better than TRMM 3B42V7; (2) IMERG V05 showed the best performance in SPI-1 (one-month SPI) estimations compared with CMORPH-CRT and TRMM 3B42V7; (3) SPI-1 was more suitable for drought monitoring than AWD in the Xiang River Basin, because its high R-values and low root mean square error (RMSE) compared with the corresponding index based on in situ observations; (4) drought conditions in 2015 were apparently more severe than that in 2016 and 2017, with the driest area mainly distributed in the southwest part of the Xiang River Basin.

We introduce the concepts of ðn; dÞ-injective and ðn; dÞ-flat as generalizations of injective, flat modules and homological dimensions, and use them to characterize right n-coherent rings and right (weak) ðn; dÞ-rings in various way. Some known results can be obtained as corollaries. ORDER REPRINTSfinite n-presentation, that is, there is an exact sequence of right R-moduleswhere each F i is a finitely generated free, equivalently projective, right R-module. Clearly every finitely generated projective module is n-presented for each n. A module is 0-presented (resp. 1-presented) if and only if it is finitely generated (resp. finitely presented), and each m-presented module is n-presented for m ! n.According to Costa (1994), a ring R is called a right n-coherent ring in case every n-presented right R-module is ðn þ 1Þ-presented. It is easy to see that R is right 0coherent (resp. 1-coherent) if and only if it is right noetherian (resp. coherent), and every n-coherent ring is m-coherent for m ! n. n-coherent rings have been investigated by many authors (see Chen and Ding, 1996;Costa, 1994;Xue, 1999). For a commutative ring R Costa (1994) called R an ðn; dÞ-ring, if every n-presented R-module has projective dimension at most d. ðn; dÞ-rings stand for several known rings in case of different values of n; d.The object of this paper is to characterize right n-coherent rings and right (weak) ðn; dÞ-rings by ðn; dÞ-injective modules and ðn; dÞ-flat modules. As generalizations of injective, flat modules and homogical dimensions, in Sec. 2, we introduceðn; dÞinjective modules and ðn; dÞ-flat modules, and obtain some basic properties of right ðn; dÞ-rings. Section 3 is mainly to provide some characterizations of right n-coherent rings, which generalize those of right noetherian rings and right coherent rings. In Sec. 4, using ðn; dÞ-flat preenvelopes and ðn; dÞ-injective precover, we show that right n-coherent rings and right ðn; dÞ-rings are closely characterized by the existences of these preenvelopes and precovers. In Sec. 5, if S ! R is an almost excellent extension, we prove that R is a right (resp. weak) ðn; dÞ-ring if and only if S is a right (resp. weak) ðn; dÞ-ring. As corollories, we obtain some results proved by Xue (1994Xue ( , 1996, generalize (Liu, 1994, Theorem 3) and give an affirmative answer to the question by Xue (1996).For any R-module M, M Ã ¼ Hom Z ðM; Q=ZÞ denotes the character module of M. Let idðMÞ (resp. pdðMÞ, fdðMÞ) denote the injective (resp. projective, flat) dimension, and rDðRÞ (resp. wDðRÞ) the right globe dimension of R. (n, d)-RINGSDefinition 2.1. ð1Þ Let n; d be non-negative integers, a right module M is called ðn; dÞ-injective, if Ext dþ1 ðP; MÞ ¼ 0 for every n-presented right module P; ð2Þ Let n; d be non-negative integers and n ! 1, a right module M is called ðn; dÞ-flat, if Tor dþ1 ðM; QÞ ¼ 0 for every n-presented left module Q. The left versions of ðn; dÞinjective and ðn; dÞ-flat are defined similarly. In the following, we assume that n; d are non-negative integers, and n ! 1 in case of...

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