形式矩阵环起源于Morita Context 的研究，2003年Каравдина将Morita Context环推广到n阶形式矩阵环，2013年申请者和周毅强证明了环上的n阶形式矩阵环由一组因子系完全决定，这表明环上的n阶形式矩阵环是通常矩阵环的自然而又非常广泛的推广，具有丰富的研究内容，引起了国际同行的关注。该项目将以组合数学、环论、模论、同调理论和图论等为工具，主要从以下几个方面对环上n阶形式矩阵环进行研究：（1）因子系的刻画和好的因子系的寻找；（2）研究环上n阶形式矩阵环的结构和性质及其在编码、密码、现代通信等方面的应用；（3）研究环上n阶形式矩阵环的模范畴和同调维数；（4）研究环上n阶形式矩阵环的图结构，包括零因子图、单位图、交换图、映射图等。该项目研究将有利于环论、同调理论、图论和线性代数理论的研究方法和研究内容的丰富和发展，推进形式矩阵环在编码、密码、现代通信等方面应用。

Formal matrix ring originated from the research of Morita Context. In 2003, Каравдина generalized Morita Context ring to formal matrix ring of order n. In 2013, Tang Gaohua and Zhou Yiqiang proved that a formal matrix ring of order n over a ring is completely determined by a system of factors. This indicates that a formal matrix ring of order n over a ring is a natural and extensive generalization of usual matrix ring and it has a rich research content, which caused wide attention for the international colleagues. This project mainly studies the formal matrix rings over a ring from the following aspects by jointly using the method on combinatorics, ring theory, module theory, homological theory and graph theory: (1) characterizing the system of factors and finding suitable systems of factors; (2) studying the structure and properties of formal matrix rings over rings and exploring its applications in coding theory, cryptography, modern communication, etc.; (3) investigating the module categories and the homological dimensions of formal matrix rings over rings; (4) determining the graph structure and invariants of formal matrix rings over rings，including zero divisor graphs, unit graphs，commuting graphs，the mapping graphs. This project will be conducive to enrich and develop the research methods and content in ring theory, homological theory, graph theory and linear algebra theory. It also promotes the applications of formal matrix rings over rings in coding theory, cryptography, modern communication, etc.