To solve an equation of the first degree on a field is not a problem since we already have a model resolution. What happens to the model resolution on a ring?
By "first degree" I assume you mean a system of linear equations? If so, then for Euclidean rings there is the Hermite canonical form of a matrix, somewhat analogous to reduced row echelon form for matrices over a field. This can be used to determine solvability and to provide solutions in solvable instances.
Let A be ring, a, b elements in A. Typically, first degree equation is like: (1) ax=b.
If a not divisor zero, you have at most one solution.
If (1) has solution and a divisor of zero, then (1) has at least two solutions.
.... etc.
My favourite references are:
-P. Camion, L. Levy, H. Mann, Linear equations over commutative rings, J.Algebra 18 (1971)
-J.A. Hermida. Linear Equations over Commutative Rings and Grade theory , J. Algebra 148, (1992)
They cover existence, uniqueness &c. over a rather general commutative rings.
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