APT环上幂等阵的对角化

On Diagonalization of Idempotent Matrices over APT Rings

英文关键词:Abelian ring  APT ring  idempotent matrix.

中文摘要:

      设R是一阿贝尔环(R的所有幂等元都在中心里)，A是R上的一幂等阵.本文证明了以下结果：(a)A相抵于一对角阵当且仅当A相似于一对角阵；(b)若R是一APT(阿贝尔投射平凡)环，则A在相似变换之下可唯一地化为对角形diag{e₁, ..., e_n｝，这里e_i整除e_i+1；(c)R是APT环当且仅当R/I是APT环，这里I是环R的一幂零理想.由(a)，还证明了分离的阿贝尔正则环是APT环.

英文摘要:

      Let R be an abelian ring ( all idempotents of R lie in the center of R), and A be an idempotent matrix over R. The following statements are proved: (a). A is equivalent to a diagonal matrix if and only if A is similar to a diagonal matrix. (b). If R is an APT (abelian projectively trivial) ring, then A can be uniquely diagonalized as diag{e₁, ..., e_n} and e_i divides e_i+1. (c). R is an APT ring if and only if R/I is an APT ring, where I is a nilpotent ideal of R. By (a), we prove that a separative abelian regular ring is an APT ring.