  Several problems in algebraic geometry and coding theory over finite rings
are modeled by systems of algebraic equations. Among these problems, we have
the rank decoding problem, which is used in the construction of public-key
cryptography. In 2004, Nechaev and Mikhailov proposed two methods for solving
systems of polynomial equations over finite chain rings. These methods used
solutions over the residual field to construct all solutions step by step.
However, for some types of algebraic equations, one simply needs partial
solutions. In this paper, we combine two existing approaches to show how
Gr\"obner bases over finite chain rings can be used to solve systems of
algebraic equations over finite commutative rings. Then, we use skew
polynomials and Pl\"ucker coordinates to show that some algebraic approaches
used to solve the rank decoding problem and the MinRank problem over finite
fields can be extended to finite principal ideal rings.
