{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T15:43:24Z","timestamp":1782229404826,"version":"3.54.5"},"reference-count":21,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T00:00:00Z","timestamp":1558137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let R be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner R-hypermodules with inclusion single-valued R-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner R-hypermodules with strong single-valued R-homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued R-homomorphism     f : A \u2192 B     and     a \u2208 A    , we have     K e r ( f ) + a = a + K e r ( f ) = { x \u2208 A \u2223 f ( x ) = f ( a ) }    .<\/jats:p>","DOI":"10.3390\/sym11050687","type":"journal-article","created":{"date-parts":[[2019,5,20]],"date-time":"2019-05-20T11:05:07Z","timestamp":1558350307000},"page":"687","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Isomorphism Theorems in the Primary Categories of Krasner Hypermodules"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5178-3699","authenticated-orcid":false,"given":"Hossein","family":"Shojaei","sequence":"first","affiliation":[{"name":"School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran P.O. Box 14155-6455, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7682-0660","authenticated-orcid":false,"given":"Dario","family":"Fasino","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Computer Science and Physics, University of Udine, 33100 Udine, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,18]]},"reference":[{"key":"ref_1","unstructured":"Corsini, P. (1993). Prolegomena of Hypergroup Theory, Aviani editore. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Corsini, P., and Leoreanu-Fotea, V. (2003). Applications of Hyperstructures Theory, Kluwer Academic Publisher.","DOI":"10.1007\/978-1-4757-3714-1"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Davvaz, B. (2013). Polygroup Theory and Related Systems, World Scientific.","DOI":"10.1142\/8593"},{"key":"ref_4","unstructured":"Davvaz, B., and Leoreanu-Fotea, V. (2007). Hyperring Theory and Applications, International Academic Press."},{"key":"ref_5","unstructured":"Vougiouklis, T. (1994). Hyperstructures and Their Representations, Hadronic Press Inc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.jalgebra.2014.05.033","article-title":"Fully simple semihypergroups, transitive digraphs, and sequence A000712","volume":"415","author":"Fasino","year":"2014","journal-title":"J. Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s00009-008-0151-4","article-title":"Minimal order semihypergroups of type U on the right","volume":"5","author":"Fasino","year":"2008","journal-title":"Mediterranean J. Math."},{"key":"ref_8","first-page":"165","article-title":"Hypergroupes canoniques","volume":"2","author":"Mittas","year":"1972","journal-title":"Math. Balkanica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1155\/S0161171283000265","article-title":"A class of hyperrings and hyperfelds","volume":"6","author":"Krasner","year":"1983","journal-title":"Internat. J. Math. Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1080\/09720529.2003.10697969","article-title":"On categories of hypergroups and hypermodules","volume":"6","author":"Ameri","year":"2003","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Madanshekaf, A. (2006). Exact category of hypermodules. Internat. J. Math. Math. Sci.","DOI":"10.1155\/IJMMS\/2006\/31368"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/BF01761468","article-title":"Free and cyclic hypermodules","volume":"150","author":"Massouros","year":"1988","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_13","first-page":"2298","article-title":"Some results on categories of Krasner hypermodules","volume":"8","author":"Shojaei","year":"2016","journal-title":"J. Fundam. Appl. Sci."},{"key":"ref_14","first-page":"38","article-title":"Some unavoidable questions about Krasner hypermodules","volume":"11","author":"Shojaei","year":"2018","journal-title":"Int. J. Open Probl. Compt. Math."},{"key":"ref_15","first-page":"475","article-title":"On properties of various morphisms in the categories of general Krasner hypermodules","volume":"39","author":"Shojaei","year":"2017","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_16","first-page":"269","article-title":"Pre-semihyperadditive categories","volume":"27","author":"Shojaei","year":"2019","journal-title":"An. Sti. U. Ovid. Co-Mat."},{"key":"ref_17","first-page":"321","article-title":"Isomorphism theorems of hyperrings","volume":"35","author":"Davvaz","year":"2004","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Velrajan, M., and Asokkumar, A. (2010). Note on isomorphism theorems of hyperrings. Internat. J. Math. Math. Sci.","DOI":"10.1155\/2010\/376985"},{"key":"ref_19","first-page":"793","article-title":"Various kinds of freeness in the categories of Krasner hypermodules","volume":"16","author":"Shojaei","year":"2018","journal-title":"Int. J. Anal. Appl."},{"key":"ref_20","first-page":"133","article-title":"Various Kinds of Quotient of a Canonical Hypergroup","volume":"9","author":"Shojaei","year":"2018","journal-title":"Sigma J. Eng. Nat. Sci."},{"key":"ref_21","unstructured":"Awody, S. (2010). Category Theory, Oxford University Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/687\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:53:21Z","timestamp":1760187201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/687"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,18]]},"references-count":21,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050687"],"URL":"https:\/\/doi.org\/10.3390\/sym11050687","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,18]]}}}