{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,26]],"date-time":"2024-03-26T23:30:36Z","timestamp":1711495836120},"reference-count":21,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2018,9]]},"abstract":"<jats:p> Let [Formula: see text] be an integer. We introduce the notions of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of [Formula: see text]-gr-coherent rings in terms of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Moreover, we prove that the pairs (gr-[Formula: see text], gr-[Formula: see text]) and (gr-[Formula: see text], gr-[Formula: see text]) are duality pairs over left [Formula: see text]-coherent rings, where gr-[Formula: see text] and gr-[Formula: see text] denote the subcategories of [Formula: see text]-FP-gr-injective left [Formula: see text]-modules and [Formula: see text]-gr-flat right [Formula: see text]-modules respectively. As applications, we obtain that any graded left (respectively, right) [Formula: see text]-module admits an [Formula: see text]-FP-gr-injective (respectively, [Formula: see text]-gr-flat) cover and preenvelope. <\/jats:p>","DOI":"10.1142\/s021819671850042x","type":"journal-article","created":{"date-parts":[[2018,6,7]],"date-time":"2018-06-07T04:35:31Z","timestamp":1528346131000},"page":"959-977","source":"Crossref","is-referenced-by-count":4,"title":["Relative FP-gr-injective and gr-flat modules"],"prefix":"10.1142","volume":"28","author":[{"given":"Tiwei","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, P. R. China"}]},{"given":"Zenghui","family":"Gao","sequence":"additional","affiliation":[{"name":"College of Applied Mathematics, Chengdu University, of Information Technology, Chengdu 610225, P. R. 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