Linear equation over a ring

In algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and the solutions that one is looking for belong to a given field, commonly the real or the complex numbers. This article is devoted to the same problems where "field" is replaced by "commutative ring", or, typically "Noetherian integral domain".

In the case of a single equation, the problem splits in two parts. First, the ideal membership problem, which consists, given a non homogeneous equation

with ${\displaystyle a_{1},\ldots ,a_{k}}$ and b in a given ring R, to decide if it has a solution with ${\displaystyle x_{1},\ldots ,x_{k}}$ in R, and, if any, to provide one. This amounts to decide if b belongs to the ideal generated by the a_i. The simplest instance of this problem is, for k = 1 and b = 1, to decide if a is a unit in R.

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