In this paper, I present the properties and structures of Dedekind $\sigma$-complete directed partially ordered rings and algebras over $\mathbb{R}$. Under certain conditions, such as property P₁, chain conditions on convex ideals, and strong Archimedean, structures are isomorphic to the real number field $\mathbb{R}$ with the usual total order or finite product or subdirect product of $\mathbb{R}$.