本项目从环论、矩阵论和广义逆理论等角度研究基于(b,c)-逆的相关问题及其应用。以复矩阵的(b,c)-逆问题为导向，以环中元素的(b,c)-逆问题为线索，可以将环论、矩阵论和广义逆理论串联起来，为研究相关的理论问题提供不同的思路。以复矩阵的(b,c)-逆问题为导向的优点在于：复矩阵的研究工具丰富，如矩阵分解、矩阵指标和矩阵的秩，可以研究(b,c)-逆的存在性条件与精确表达式及满足交换性、反序律的条件。以环中元素的(b,c)-逆问题为线索的优点在于：环论的方法具有简洁、快速计算等优点，借鉴环论的方法研究复矩阵的(b,c)-逆问题是一个新的研究思路。本项目的实施将有助于丰富广义逆理论。广义逆理论在数理统计、数值分析、网络分析和图论等学科中有着重要的应用，可用于解决奇异多元正态分布、线性约束下二次最小化问题、区间线性规划、最优化、非线性方程组等问题。

This project will study the related problems with applications of the (b,c)-inverse from the perspectives of ring theory, matrix theory and generalized inverse theory. Guided by the (b,c)-inverse problem of complex matrices and the (b,c)-inverse problem of elements in rings, the relationships of ring theory, matrix theory and generalized inverse theory can be found, which provide different ideas for researching related theoretical issues. The advantages of the (b,c)-inverse problem of complex matrices are: the tools of complex matrix are abundant, such as decomposition, index and rank can be used to investigate the existence conditions and explicit expressions of the (b,c)-inverse and the conditions of commutability and reverse order law. The advantages of the (b,c)-inverse problem in a ring are: the method of the ring theory has the advantages of simplicity and fast calculation. Using the method of ring theory to study the (b,c)-inverse problems of complex matrices is a new research idea. The implementation of this project will help enrich the theory of generalized inverse.Theory of generalized inverse has important applications in mathematical statistics,numerical analysis,network analysis,graph theory and so on. This theory can be used to solve the problems of singular multivariate normal distribution, quadratic minimization problems restricted by linear constraints, interval linear programming, optimization, nonlinear equations, etc.