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In this paper, we first obtain equivalent conditions for a module over a Prüfer domain and derive properties of Dedekind modules over such a domain. We then obtain equivalent conditions for a finitely generated Dedekind module over an integrally closed ring. Finally, we characterize multiplication Dedekind modules.

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An element of a ring R is called nil-clean if it is the sum of an idempotent and a nilpotent element. A ring is called nil-clean if each of its elements is nil-clean. S. Breaz et al. in [1] proved their main result that the matrix ring Mn(F ) over a field F is nil-clean if and only if F ∼ = F2, where F2 is the field of two elements. M. T. Koşan et al. generalized this result to a division ring. In this paper, we show that the n × n matrix ring over a principal ideal domain R is a nil-clean ring if and only if R is isomorphic to F2. Also, we show that the same result is true for the 2 × 2 matrix ring over an integral domain R. As a consequence, we show that for a commutative ring R, if M2(R) is a nil-clean ring, then dimR = 0 and charR/J(R) = 2.