First-order implicit linear difference equation over finite commutative rings with identity

Source abstract

The paper studies an implicit first-order linear difference equation $BX_{n+1}=AX_n+F_n,\quad n=0,1,2,\ldots$ over a finite commutative ring $R$ with identity, that is, an equation with a noninvertible element $B$ of the ring $R$. In contrast to the classical (explicit) linear difference equation, an implicit linear difference equation over the finite ring $R$ may have no solutions, and may also have infinitely many solutions. Since any finite commutative ring with identity is isomorphic to a finite direct sum of local commutative rings with identity, the equation decomposes into a system of equations over local finite commutative rings with identity. It is shown that the condition that the ideal $(A,B)$ generated by the elements $A,B\in R$ coincides with $R$ is necessary and sufficient for the existence of a finite number of solutions of this equation; the number of solutions in the case of their existence is counted and a formula for the general solution is provided. The condition $(A,B)\ne R$ is a necessary and sufficient condition for the existence of an infinite number of solutions of the corresponding homogeneous equation $BX_{n+1}=AX_n,\quad n=0,1,2,\ldots$. It is also established that in the case $(A,B)\ne R$ the condition $F_n\in (A,B),\quad n=0,1,2,\ldots$ is necessary for the solvability of the nonhomogeneous implicit linear difference equation, but it is not sufficient, as examples show. Under the additional restriction that $(A,B)$ is a proper principal ideal of the ring $R$, this condition is also sufficient for the existence of a solution of the considered nonhomogeneous equation; in this situation the equation has infinitely many solutions. Under the assumption that $(A,B)$ is a principal ideal, first a criterion for the existence of a solution is proved in the case of a local finite commutative ring with identity, and then in the case of an arbitrary finite commutative ring with identity. As in Fredholm theory, it is shown that if the corresponding homogeneous equation has only the trivial solution, then the studied nonhomogeneous equation has a unique solution. The work of the proved theorems is demonstrated by concrete examples.