Let R be a discrete valuation ring with quotient field F. Let K/F be a finite Galois extension with automorphism group G, and let S be the integral closure of R in K. Assume that S/R is (at most) tamely ramified. Let f : G × G → K^× be a 2-cocycle which takes values in S. This thesis examines properties of the R-order A_f := Σ_σ∈G Sx_σ in the crossed product algebra Σ_f := Σ _σ∈G Kx_σ. Because f may take values in S that are nonunits, we say that A_f is a weak crossed product order in Σ_f .

Let H = {σ ∈ G|f(σ,σ ^-1) is a unit of S} be the so-called inertial subgroup for A_f. In the case where S is a DVR, it is shown that the Jacobson radical of A_f is given by Rad(A_f) = Σ _h∈H π_SSx _h + Σ_σ∈H Sx_σ, where π_S generates the prime ideal of S. The order A_f is hereditary if and only if v_S(f( σⁱ, σ^j)) ∈ {0, 1} for every i and j. There is a partial ordering on G/H defined by g₁ H ≤ g₂H if and only if f(g₁, g ₁^-1g₂) is a unit of S. When A_f is hereditary, this ordering is a total ordering, and G/H is a cyclic group which is generated by the minimal nonidentity coset of G/H. For hereditary crossed product orders, the inertia group (in the sense of the theory of Dedekind domains) is a subgroup of H. The hereditary crossed product orders are also seen to be maximal among crossed product orders in Σ _f. Under an additional hypothesis, these results can be used to determine the hereditarity of A_f locally when S is a semilocal Dedekind domain.