{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:40:04Z","timestamp":1759639204880,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,3]],"date-time":"2025-10-03T00:00:00Z","timestamp":1759449600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004410","name":"TUBITAK","doi-asserted-by":"crossref","award":["123F236"],"award-info":[{"award-number":["123F236"]}],"id":[{"id":"10.13039\/501100004410","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004329","name":"Slovenian Research and Innovation Agency","doi-asserted-by":"crossref","award":["P2-0103"],"award-info":[{"award-number":["P2-0103"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we introduce ps-supplement submodules as a kind of supplements. Specifically, a submodule V of a module M is referred to as a ps-supplement if there exists another submodule U\u2286M such that M=U+V and the intersection U\u2229V is projective semisimple. We demonstrate that, over an arbitrary ring, the collection PS of short exact sequences of the form 0\u27f6M\u27f6\u03c8N\u27f6\u03d5K\u27f60, where Im(\u03c8) is a ps-supplement in N, constitutes a proper class. In addition, we explore several homological aspects of this class, including the symmetry of the relations between the modules. Finally, we provide that a ring R is a left SI-ring with essential socle if and only if every left R-module is ps-supplemented.<\/jats:p>","DOI":"10.3390\/sym17101642","type":"journal-article","created":{"date-parts":[[2025,10,3]],"date-time":"2025-10-03T07:57:27Z","timestamp":1759478247000},"page":"1642","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On ps-Supplement Submodules"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7032-6485","authenticated-orcid":false,"given":"\u0130rfan","family":"Soydan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Institute of Science, Amasya University, 05100 Amasya, Turkey"}]},{"given":"Erg\u00fcl","family":"T\u00fcrkmen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Art and Science, Amasya University, 05100 Amasya, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3871-217X","authenticated-orcid":false,"given":"Hashem","family":"Bordbar","sequence":"additional","affiliation":[{"name":"Center for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,3]]},"reference":[{"key":"ref_1","first-page":"449","article-title":"Small supplements, weak supplements and proper classes","volume":"45","author":"Alizade","year":"2016","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1080\/00927872.2012.699567","article-title":"The proper class generated by weak supplements","volume":"42","author":"Alizade","year":"2014","journal-title":"Commun. Algebra"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"157","DOI":"10.4171\/rsmup\/124-10","article-title":"Rad-supplemented modules","volume":"124","author":"Mermut","year":"2010","journal-title":"Rend. Semin. Mat. Univ. Padova"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Demirci, Y.M., and T\u00fcrkmen, E. (2022). WSA-Supplements and Proper Clases. Mathematics, 10.","DOI":"10.3390\/math10162964"},{"key":"ref_5","first-page":"147","article-title":"Sa-supplement submodules","volume":"58","year":"2021","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2833","DOI":"10.3906\/mat-1907-107","article-title":"Extended S-supplement submodules","volume":"43","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1134\/S0001434613030073","article-title":"On Supplementation and Generalized Projective Modules","volume":"93","author":"Kamal","year":"2013","journal-title":"Math. Notes"},{"key":"ref_8","first-page":"127","article-title":"On Principally Lifting Modules","volume":"2","author":"Kamal","year":"2007","journal-title":"Int. Electron. J. Algebra"},{"key":"ref_9","first-page":"671","article-title":"Modules closed full large extensions of cyclic modules are summands","volume":"43","author":"Mehany","year":"2020","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"66","DOI":"10.2307\/1970093","article-title":"A note on homology in categories","volume":"69","author":"Buchsbaum","year":"1959","journal-title":"Ann. Math."},{"key":"ref_11","unstructured":"Clark, J., Lomp, C., Vanaja, N., and Wisbauer, R. (2006). Lifting Modules. Supplements and Projectivity in Module Theory, Birkh\u00e4user."},{"key":"ref_12","unstructured":"Wisbauer, R. (1960). Foundations of Module and Ring Theory, Gordon and Breach Science Publishers."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1142\/S1005386707000065","article-title":"\u03b4-lifting and \u03b4-supplemented modules","volume":"14","year":"2007","journal-title":"Algebra Colloq."},{"key":"ref_14","first-page":"1051","article-title":"Small-essential submodules and morita duality","volume":"35","author":"Zhou","year":"2011","journal-title":"Southeast Asian Bull. Math."},{"key":"ref_15","first-page":"473","article-title":"ss-supplemented modules","volume":"69","author":"Kaynar","year":"2020","journal-title":"Commun. Fac. Sci. Univ. Ank. Ser. Math. Stat."},{"key":"ref_16","first-page":"193","article-title":"\u03b4ss-supplemented modules and rings","volume":"28","year":"2020","journal-title":"Analele Stiintifice Ale Univ. Ovidius Constanta"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s10011-000-0305-9","article-title":"Generalizations of perfect, semiperfect, and semiregular rings","volume":"7","author":"Zhou","year":"2000","journal-title":"Algebra Colloq."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"MacLane, S. (1963). Homology, Springer.","DOI":"10.1007\/978-3-642-62029-4"},{"key":"ref_19","first-page":"121","article-title":"Purity and subfunctors of the identity in Topics in abelian groups","volume":"3","author":"Nunke","year":"1963","journal-title":"Proc. Symp. New Mex. State Univ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1090\/S0002-9947-1971-0274511-2","article-title":"Endomorphism rings of projective modules","volume":"155","author":"Ware","year":"1971","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Jain, S.K., Srivastava, A.K., and Tuganbaev, A.A. (2012). Cyclic Modules and the Structure of Rings, Oxford University Press. Oxford Mathematical Monographs.","DOI":"10.1093\/acprof:oso\/9780199664511.001.0001"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s11253-010-0344-4","article-title":"On generalization of \u2295-cofinitely supplemented modules","volume":"62","author":"Pancar","year":"2010","journal-title":"Ukr. Math. J."},{"key":"ref_23","unstructured":"Mermut, E. (2004). Homological Approach to Complements and Supplements. [Ph.D. Thesis, Dokuz Eyl\u00fcl University]."},{"key":"ref_24","unstructured":"Kasch, F. (1982). Modules and Rings, Academic Press."},{"key":"ref_25","first-page":"41","article-title":"Some relations between semi-Artinian rings and Goldie\u2019s torsion theory","volume":"23","author":"Raggi","year":"1983","journal-title":"An. Inst. Mat. Univ. Nac. Auton."},{"key":"ref_26","unstructured":"Misina, A.P., and Skornjakov, L.A. (1960). Abelian Groups and Modules, American Mathematical Society."},{"key":"ref_27","unstructured":"Dung, N.V., Huynh, D., Smith, P.F., and Wisbauer, R. (1994). Extending Modules, Longman."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4229","DOI":"10.1080\/00927870500279001","article-title":"Soc-injective rings and modules","volume":"33","author":"Amin","year":"2005","journal-title":"Commun. Algebra"},{"key":"ref_29","first-page":"117","article-title":"Generalized V-rings and von Neumann regular rings","volume":"72","author":"Bacella","year":"1984","journal-title":"Rend. Semin. Mat. Univ. Padova"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/S0021-8693(02)00095-9","article-title":"Structure of noetherian SI rings","volume":"254","author":"Huynh","year":"2001","journal-title":"J. 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