Since on any commutative algebra $R$ over ring $S$ we have module of Kahler differentials $(\Omega_{R/S},d)$ which extends to the algebraic de-Rham complex $(\Omega^\bullet,d),$ it is natural to define connection on $R$-module $P$ as any $S$-linear map $\nabla:P\to\Omega_{R/S}\otimes P$ such that $$\nabla(rp)=dr\otimes p + r\nabla(p).$$ As in geometry we can extend $\nabla$ to $d^\nabla:\Omega^\bullet\otimes P\to\Omega^{\bullet+1}\otimes P$ and define its curvature $F^\nabla.$

I look for some source which analyze such approach. The only thing I found are Sardanashvilys Lectures on Differential Geometry of Modules and Rings which define connection via jet bundle, but I guess it is equivalent definition. However this source is from 2009 and in 1995 Dubois-Violette and Michor wrote Connections on central bimodules in which they investigate connections in noncommutative case. They also refer to Koszules Lectures on fibre bundles and differentials, but Koszul does not write anything about kahler differentials.

Question.Is there any source which analyze connections from purely commutative algebraic point of view?Or maybe there is no point for such approach because there is no interesting connections beyond algebra of smooth functions on smooth manifold?

**Some footnote.**
In commutative algebra connections can be defined even for any dga as in ncatlab, but there is no reference so again I do not know where such case is investigated.

notwhat you are looking for, but there is Bertram's monograph "Differential Geometry over General Base Fields and Rings" (2005) and Betram, Glöckner and Neeb's "Differential Calculus, Manifolds, and Lie Groups over Arbitrary Infinite Fields" (2003). I don't know if their approach can be of any interest to you. $\endgroup$ – M.G. Dec 12 '16 at 17:27Local Cohomological Dimension of Algebraic Varietiesby Arthur Ogus, (Annals of Mathematics Second Series, Vol. 98, No. 2 (Sep., 1973), pp. 327-365). Not sure if that's what you are looking for but it may be relevant. $\endgroup$ – Mahdi Majidi-Zolbanin Dec 13 '16 at 19:34