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More characterizations of Dedekind domains and V-rings

In this paper, cyclic c−injective modules are introduced and investigated. It is shown that a commutative Noetherian domain is Dedekind if and only if every simple module is cyclic c−injective. Finally, it is shown that injectivity, principal injectivity, mininjectivity, and simple injectivity are all equal to characterize right V-rings, right GV-rings, right pV-rings, and WV-rings. | Turk J Math (2017) 41: 33 – 42 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1504-13 Research Article More characterizations of Dedekind domains and V-rings Thuat Van DO1 , Hai Dinh HOANG2 , Samruam BAUPRADIST3,∗ Department of Science and Technology, Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam 2 International Cooperation Office, Hong Duc University, Thanh Hoa City, Thanh Hoa, Vietnam 3 The corresponding author, Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, Thailand 1 Received: 10.04.2015 • Accepted/Published Online: 09.03.2016 • Final Version: 16.01.2017 Abstract: In this paper, cyclic c− injective modules are introduced and investigated. It is shown that a commutative Noetherian domain is Dedekind if and only if every simple module is cyclic c− injective. Finally, it is shown that injectivity, principal injectivity, mininjectivity, and simple injectivity are all equal to characterize right V-rings, right GV-rings, right pV-rings, and WV-rings. Key words: c− Injective modules, Dedekind domains, right V-rings, right GV-rings, WV-rings 1. Introduction Throughout this paper, all rings are associative with identity and all modules are unitary (unless otherwise stated). Recall that an integral domain R that is not a field is called a Dedekind domain if every nonzero proper ideal factors into primes. This is equivalent to R being an integrally closed, Noetherian domain with Krull dimension one (i.e. every nonzero prime ideal is maximal). Mermut et al. [10] showed that a commutative Noetherian domain is Dedekind if and only if every simple module is c− injective. According to Baccella [1], a ring R is called a right V-ring (resp. right GV-ring) if every simple (resp. simple singular) right R− module is RR − injective, i.e. injective. According to L´opez-Permouth et al. [9], a ring R is called a right pV-ring provided every simple right R− .