{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:36:18Z","timestamp":1753886178153,"version":"3.41.2"},"reference-count":28,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2009,12,22]],"date-time":"2009-12-22T00:00:00Z","timestamp":1261440000000},"content-version":"vor","delay-in-days":355,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>The so\u2010called Pr\u00fcfer <jats:italic>\u03c5<\/jats:italic>\u2010multiplication domains (P<jats:italic>\u03c5<\/jats:italic>MDs) are usually defined as domains whose finitely generated nonzero ideals are <jats:italic>t<\/jats:italic>\u2010invertible. These domains generalize Pr\u00fcfer domains and Krull domains. The P<jats:italic>\u03c5<\/jats:italic>MDs are relatively obscure compared to their very well\u2010known special cases. One of the reasons could be that the study of P<jats:italic>\u03c5<\/jats:italic>MDs uses the jargon of star operations, such as the <jats:italic>\u03c5<\/jats:italic>\u2010operation and the <jats:italic>t<\/jats:italic>\u2010operation. In this paper, we provide characterizations of and basic results on P<jats:italic>\u03c5<\/jats:italic>MDs and related notions without star operations.<\/jats:p>","DOI":"10.1155\/2009\/349010","type":"journal-article","created":{"date-parts":[[2009,12,22]],"date-time":"2009-12-22T15:30:17Z","timestamp":1261495817000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A \u201c<i>\u03c5<\/i>\u2010Operation Free\u201d Approach to Pr\u00fcfer <i>\u03c5<\/i>\u2010Multiplication Domains"],"prefix":"10.1155","volume":"2009","author":[{"given":"Marco","family":"Fontana","sequence":"first","affiliation":[]},{"given":"Muhammad","family":"Zafrullah","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2009,12,22]]},"reference":[{"key":"e_1_2_4_1_2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1967-065-8"},{"key":"e_1_2_4_2_2","doi-asserted-by":"publisher","DOI":"10.1080\/00927870802502688"},{"key":"e_1_2_4_3_2","series-title":"Proceedings of the Fez Conference","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1515\/9783110213188.247","volume-title":"Commutative Algebra and Its Applications","author":"Halter-Koch F.","year":"2009"},{"volume-title":"Commutative Algebra: Noetherian and Non-Noetherian Perspectives","year":"2009","author":"Fontana M.","key":"e_1_2_4_4_2"},{"key":"e_1_2_4_5_2","doi-asserted-by":"crossref","first-page":"133","DOI":"10.24033\/bsmf.1329","article-title":"Sur la th\u00e9orie de la divisibilit\u00e9","volume":"69","author":"Dieudonn\u00e9 J.","year":"1941","journal-title":"Bulletin de la Soci\u00e9t\u00e9 Math\u00e9matique de France"},{"volume-title":"Multiplicative Ideal Theory","year":"1972","author":"Gilmer R.","key":"e_1_2_4_6_2"},{"key":"e_1_2_4_7_2","doi-asserted-by":"publisher","DOI":"10.1080\/00927878508823247"},{"key":"e_1_2_4_8_2","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1029005623"},{"key":"e_1_2_4_9_2","series-title":"Mathematics and Its Applications","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/978-1-4757-3180-4_20","volume-title":"Non-Noetherian Commutative Ring Theory","author":"Zafrullah M.","year":"2000"},{"key":"e_1_2_4_10_2","unstructured":"ZafrullahM. 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