We present, in the same vein as in [20] and [21], some results of the so-called "Smooth (or $\mathcal{ C}^\infty$) Commutative Algebra", a version of Commutative Algebra of $\mathcal{ C}^{ \infty}-$rings instead of ordinary commutative unital rings, looking for similar results to those one finds in the latter, and expanding some others presented in [20]. We give an explicit description of an adjunction between the categories $\mathcal{ C}^\infty{ \rm Rng}$ and ${ \rm CRing}$, in order to study this "bridge". We present and prove many properties of the analog of the radical of an ideal of a ring (namely, the $\mathcal{ C}^\infty$-radical of an ideal), saturation (which we define as "smooth saturation", inspired by [13]), rings of fractions ($\mathcal{ C}^\infty$-rings of fractions, defined first by I. Moerdijk and G. Reyes in [20]), local rings (local $\mathcal{ C}^\infty$-rings), reduced rings ($\mathcal{ C}^\infty$-reduced $\mathcal{ C}^\infty$-rings) and others. We also state and prove new results, such as ad hoc "Separation Theorems", similar to the ones we find in Commutative Algebra, and a stronger version (Theorem 6) of the Theorem 1.4 of [20], characterizing every $\mathcal{ C}^\infty$-ring of fractions. We describe the fundamental concepts of Order Theory for $\mathcal{ C}^\infty$-rings, proving that every $\mathcal{ C}^\infty$-ring is semi-real, and we prove an important result on the strong interplay between the smooth Zariski spectrum and the real smooth spectrum of a $\mathcal{ C}^\infty$-ring.
Categories: math.AC