Implicit linear difference equations over finite commutative rings of order p^2 with identity
Abstract
It is known that, up to isomorphism, there are exactly four finite commutative rings with identity, whose order is equal to $p^2$, where p is a prime number. Namely, these rings are the residue class ring modulo $p^2$, the direct sum of two residue class rings $\mathbb{Z}_p$ modulo $p$, the field of order $p^2$ and the ring $\mathcal{S}_p = \mathbb{Z}_p[t]/(t^2)$. Recently, a solvability criterion was established for the first-order linear difference equation over the residue class ring modulo $m \ge 2$. Considering this, it appears necessary to solve the solvability problem for the linear difference equation over the ring $\mathcal{S}_p$ of order $p^2$.
This paper investigates first-order implicit linear difference equations over the ring $\mathcal{S}_p$. The paper presents the solvability criterion for the mentioned equation over this ring. In addition, the obtained results describe both the number of solutions and the form of the general solution of this equation. Analogous results were obtained for the initial problem over the ring $\mathcal{S}_p$. In particular, it was established that, unlike in the case of an integral domain, the initial problem over the ring $\mathcal{S}_p$ may have infinitely many solutions. Moreover, if it has a finite number of solutions, then the solution of this initial problem is unique. We obtain several corollaries of the solvability criterion for the implicit linear difference equation over the ring $\mathcal{S}_p$. In particular, as in Fredholm theory, we show that if a homogeneous equation, which corresponds to the non-homogeneous equation, has only the trivial solution, then the non-homogeneous equation, which is being investigated, has a unique solution.
The article includes an example demonstrating the application of the obtained theoretical results to solving a certain equation over the ring $\mathcal{S}_p$ and the corresponding initial problem.
The results may be applied to further studies of linear difference equations over finite rings, and also to the general theory of discrete dynamical systems.
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References
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