{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:07Z","timestamp":1760145067070,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T00:00:00Z","timestamp":1718668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The Scientific and Technological Research Council of Turkey (TUBITAK)","award":["123F236","P2-0103"],"award-info":[{"award-number":["123F236","P2-0103"]}]},{"name":"Slovenian Research and Innovation Agency","award":["123F236","P2-0103"],"award-info":[{"award-number":["123F236","P2-0103"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study is motivated by the recently published papers on normal injective and normal projective hypermodules. We provide a new characterization of the normal injective and normal projective hypermodules by using the splitting of the short exact sequences of hypermodules. After presenting some of their fundamental properties, we show that if a hypermodule is normal projective, then every exact sequence ending with it is splitting. Moreover, if a hypermodule is normal injective, then every exact sequence starting with it is splitting as well. Finally, we investigate the relationships between semisimple, simple, normal injective, and normal projective hypermodules.<\/jats:p>","DOI":"10.3390\/axioms13060410","type":"journal-article","created":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T03:42:06Z","timestamp":1718682126000},"page":"410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Characterization of Normal Injective and Normal Projective Hypermodules"],"prefix":"10.3390","volume":"13","author":[{"given":"Erg\u00fcl","family":"T\u00fcrkmen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Art and Science, Amasya University, 05100 Ipekkoy, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7900-0529","authenticated-orcid":false,"given":"Burcu N\u0130\u015fanc\u0131","family":"T\u00fcrkmen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Art and Science, Amasya University, 05100 Ipekkoy, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3871-217X","authenticated-orcid":false,"given":"Hashem","family":"Bordbar","sequence":"additional","affiliation":[{"name":"Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,18]]},"reference":[{"key":"ref_1","unstructured":"Marty, F. (2021, January 20\u201326). Sur une generalization de la notion de group. Proceedings of the 8th Congress Mathematics, Scandinaves, Stockholm."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1155\/S0161171283000265","article-title":"A class of hyperrings and hyperfields","volume":"6","author":"Krasner","year":"1983","journal-title":"Int. J. Math. Sci."},{"key":"ref_3","first-page":"711","article-title":"Sugli iperanelli moltiplicativi","volume":"2","author":"Rota","year":"1982","journal-title":"Rend. Mat. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Vougiouklis, T. (1991). The fundamental relation in hyperrings. The general hyperfield. Algebraic Hyperstructures, and Applications (Xanthi, 1990), World Scientific Publishing.","DOI":"10.1142\/1274"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/BF01761468","article-title":"Free and cyclic hypermodules","volume":"159","author":"Massouros","year":"1988","journal-title":"Ann. Math. Pure Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Massouros, C.G., and Massouros, C. (2020). Hypercompositional Algebra, Computer Sciences and Geometry. Mathematics, 8.","DOI":"10.3390\/math8081338"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Massouros, C., and Massouros, C.G. (2021). An Overview of the Foundations of the Hypergroup Theory. Mathematics, 9.","DOI":"10.3390\/math9091014"},{"key":"ref_8","unstructured":"Vougiouklis, T. (1994). Hyperstructures and Their Representations, Hadronic Press, Inc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2150186","DOI":"10.1142\/S0219498821501863","article-title":"Projective and Injective Krasner Hypermodules","volume":"20","author":"Ameri","year":"2021","journal-title":"J. Algebra Its Appl."},{"key":"ref_10","first-page":"121","article-title":"On Categories of Hypergroups and Hypermodules","volume":"2","author":"Ameri","year":"2003","journal-title":"J. Discret. Math. Sci.-Cryptogr."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bordbar, H., and Cristea, I. (2021). About the Normal Projectivity and Injectivity of Krasner Hypermodules. Axioms, 10.","DOI":"10.3390\/axioms10020083"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ni\u015fanc\u0131 T\u00fcrkmen, B., Bordbar, H., and Cristea, I. (2022). Supplements Related to Normal \u03c0-Projective Hypermodules. Mathematics, 10.","DOI":"10.3390\/math10111945"},{"key":"ref_13","first-page":"57","article-title":"Divisible Hypermodules","volume":"30","author":"Bordbar","year":"2022","journal-title":"Analele Univ. Ovidius Constanta"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Bordbar, H. (2023). Torsion Elements and Torsionable Hypermodules. Mathematics, 11.","DOI":"10.3390\/math11214525"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2250114","DOI":"10.1142\/S0219498822501146","article-title":"Projectivity relative to closed (neat) submodules","volume":"21","author":"Durgun","year":"2022","journal-title":"J. Algebra Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1550083","DOI":"10.1142\/S0219498815500838","article-title":"Rings whose modules have maximal or minimal subprojectivity domain","volume":"14","author":"Durgun","year":"2015","journal-title":"J. Algebra Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2050091","DOI":"10.1142\/S0219498820500917","article-title":"Subprojectivity domains of pure-projective modules","volume":"19","author":"Durgun","year":"2022","journal-title":"J. Algebra Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1017\/S001708951000025X","article-title":"Poor modules: The opposite of injectivity","volume":"52","author":"Alahmadi","year":"2010","journal-title":"Glasg. Math. J."},{"key":"ref_19","first-page":"1547","article-title":"The opposite of injectivity by proper classes","volume":"46","author":"Durgun","year":"2022","journal-title":"Quaest. Math."},{"key":"ref_20","unstructured":"Corsini, P. (1993). Prolegomena of Hypergroup Theory, Aviani Editore. [2nd ed.]."},{"key":"ref_21","unstructured":"Davvaz, B., and Leoreanu-Fotea, V. (2007). Hyperring Theory and Applications, International Academic Press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1155\/S0161171285000813","article-title":"Methods of constructing hyperfields","volume":"8","author":"Massouros","year":"1985","journal-title":"Int. J. Math. Sci."},{"key":"ref_23","first-page":"5","article-title":"Small Subhypermodules and Their Applications","volume":"4","author":"Talaee","year":"2013","journal-title":"Rom. J. Math. Comput. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"T\u00fcrkmen, E., Ni\u015fanc\u0131 T\u00fcrkmen, B., and Bordbar, H. (2024). A Hyperstructural Approach to Semisimplicity. Axioms, 13.","DOI":"10.3390\/axioms13020081"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/410\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:00:21Z","timestamp":1760108421000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,18]]},"references-count":24,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060410"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060410","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,6,18]]}}}