## 1. Introduction

^{th}Century. W. Klingenberg [1] followed the study of incidence structures of D. Barbilian [2] and introduced Klingenberg spaces (initially projective spaces with homomorphisms). The definition of a PKS of general finite dimension was presented by H. H. Lück in [3]. These topics were also developed by J. C. Ferrar and F. D. Veldkamp [4,5,6] and P. Y. Bacon (e.g., [7]). Projective geometry is also related to the theory of geodesic mappings (e.g., [8]). F. Machala [9] introduced projective Klingenberg spaces over local rings. The arithmetical fundament of such spaces is a free finite-dimensional A-module over a local ring A (A-space in the sense of B.R. McDonald [10]). Subspaces (points, lines, hyperplanes, etc.) of a PKS over a ring A are subsets of points whose homomorphic images are subspaces of the respective dimension of the projective space over $A/Rad\left(A\right)$ mentioned above. Thus, PKS may be also treated as a projective space over a ring in the sense of Bingen. In the case of PKS over certain local rings (plural algebras,11]), we may study in more detail the structure of PKS, and we can find some special properties (for example, we introduce the degree of neighborhood). By this apparatus, we may describe not only subspaces of PKS, but also such sets of points (submodules of PKS) whose arithmetical representatives belong to a general submodule of the arithmetical fundament of PKS.

**Definition**

**1.**

**Remark**

**1.**

**Theorem**

**1.**

- 1.
- Any linearly-independent system of elements of M may be completed to a basis of M,
- 2.
- A submodule of M is a free submodule if and only if it is a direct summand of M.
- 3.
- An intersection of two A-subspaces of M is an A-subspace of M if and only if the sum of them is an A-subspace of M.

**Remark**

**2.**

**Theorem**

**2.**

- 1.
- ${\mathcal{B}}_{0}\cup \cdots \cup {\mathcal{B}}_{r-1}\cup {\mathcal{B}}_{r}$ is a basis of M,
- 2.
- ${\eta}^{m-r}{\mathcal{B}}_{0}\cup {\eta}^{m-r+1}{\mathcal{B}}_{1}\cup \cdots \cup {\eta}^{m-1}{\mathcal{B}}_{r-1}$ is a set of generators of K.

**Proof.**

## 2. Projective Klingenberg Spaces

**Definition**

**2.**

- 1.
- If ${X}_{1},\cdots ,{X}_{k},\phantom{\rule{4pt}{0ex}}1\le k\le n,$ are points in P such that $\mu \left({X}_{1}\right),\cdots ,\mu \left({X}_{k}\right)$ are independent in ${P}_{0}$, then there exists a hyperplane $\mathcal{H}$ in H such that ${X}_{1},\cdots ,{X}_{k}\phantom{\rule{4pt}{0ex}}I\phantom{\rule{4pt}{0ex}}\mathcal{H}$. This hyperplane is unique if $k=n$.
- 2.
- this condition is the dual of one.
- 3.
- If ${X}_{1},\cdots ,{X}_{n-1}\in P$ and ${\mathcal{H}}_{1},{\mathcal{H}}_{2}\in H$ are such that $\mu \left({X}_{1}\right),\cdots ,\mu \left({X}_{n-1}\right)\in {P}_{0}$, as well as $\mu \left({H}_{1}\right),$ $\mu \left({H}_{2}\right)\in {H}_{0}$ are independent and ${P}_{1},\cdots ,{P}_{n-1}\phantom{\rule{4pt}{0ex}}I\phantom{\rule{4pt}{0ex}}{\mathcal{H}}_{1},{\mathcal{H}}_{2}$, then: $(Y\phantom{\rule{0.166667em}{0ex}}I\phantom{\rule{0.166667em}{0ex}}{\mathcal{H}}_{1},{\mathcal{H}}_{2}\phantom{\rule{0.166667em}{0ex}}\wedge \phantom{\rule{0.166667em}{0ex}}{P}_{1},\cdots ,{P}_{n-1}\phantom{\rule{0.166667em}{0ex}}I\phantom{\rule{0.166667em}{0ex}}\mathcal{H})\Rightarrow Y\phantom{\rule{0.166667em}{0ex}}I\phantom{\rule{0.166667em}{0ex}}\mathcal{H}.$

**Definition**

**3.**

- 1.
- points in ${\mathcal{P}}_{A}$ are just all submodules $\left[x\right]\subseteq M$ such that $\langle x\rangle $ is a nonzero element of $\overline{M}$,
- 2.
- hyperplanes in ${\mathcal{P}}_{A}$ are just all submodules $[{u}_{1},{u}_{2}\cdots {u}_{n}]\subseteq M$ such that $\langle {u}_{1}\rangle ,\langle {u}_{2}\rangle ,\cdots ,\langle {u}_{n}\rangle $ are linearly-independent elements of $\overline{M}$,
- 3.
- the incidence relation is an inclusion,

**Remark**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**3.**

**Definition**

**6.**

**Definition**

**7.**

**Corollary**

**1.**

**Lemma**

**1.**

**Proof.**

**Definition**

**8.**

**Theorem**

**4.**

**Proof.**

## Funding

## Acknowledgments

## Conflicts of Interest

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