Rank of formal matrix. System of formal linear equations. Zero divisors

In this paper, we present the notion of the formal rank, i.e., the rank of a formal matrix over an arbitrary commutative ring, and some its general properties. Next, we introduce the notion of systems of formal linear equations and give necessary and sufficient conditions for the existence of a solution of homogenous systems of formal linear equations. In Section 2, we show that Cramer's rule is still valid for systems of formal linear equations. Finally, in Section 3, we establish the condition under which a formal matrix is a left or right zero divisor.