关键词: 局部环, 不可逆元, 理想格, 同调性质, 置换环, 矩阵环, 对角化

Abstract: Firstly，the homological properties of K₀ groups and K₁ groups were studied on the basis of localization from the nilpotency of irreversible elements in rings and the chain condition of ideal lattices of rings， respectively. Then， starting from the freedom of projective modules， the conditions of constructing local rings were given， and the determinant mapping properties of matrix rings and the diagonalization of idempotent matrices were studied. Finally，as an application，some results of localization of SF ring were given.

Key words: local ring, irreversible element, ideal lattice, homological property, permutation ring, matrix ring, diagonalization