Dedekind Modules

Some Remarks on Dedekind Modules

“…(iii) If M is a D 1 -module, then by [2,Lemma 3.5], M is isomorphic to an Rsubmodule of K. Conversely, suppose that M is isomorphic to an R-submodule of the quotient field K. Since R is an integral domain, M is a D 1 -module.…”

“…By [2,Lemma 3.5], M is isomorphic to a submodule L of K. (ii)⇒(iii) Let p be a maximal ideal of R and let m1 s1 , m2 s2 be nonzero elements of M p . Put N = m1 s1 , m2 s2 .…”

Euclidean modules

“…In 1996, Naoum and Al-Alwan [1] generalized the notion of Dedekind domain to the case of modules and introduced the notion of Dedekind modules. In 2005, Alkan, Sarac, and Tiras [2] gave certain characterizations of Dedekind modules and Dedekind domains. In 2002, Rahimi [3] extended the notion of Euclidean ring to Euclidean modules over commutative rings and showed that a torsion-free cyclic module over a commutative ring R with identity is Euclidean if and only if R is a Euclidean ring.…”

“…From Proposition 4.1 and Theorem 4.5, we know that M is a projective module and a Dedekind module. The rest of the proof is follows from Theorems 3.12 and 3.14 in [2].…”

Integrally Closed Modules

“…Alkan , B. Sarac and Y.Tyras( [1]) introduced the concept of integral closedness for modules, a generalization of the concept of integral closedness for a ring.…”