除环上全阵环的直积

Direct Product of Matrix Rings over Division Rings

本文给出了一个环为除环上全阵环的直积和一个环为有单位元的单环之直积的充要条件。

Let R be a ring. We prove:1. R≌multiply from i∈1 R_i, where R_i is a simple ring with identity, iff there exists a set {f_α}α∈J of central idempotent elements of R satisfying (1) for any {γ_α}α∈J (?) R there exists only a γ∈R such that e_αr=e_αγ_α for each α∈J; (2) every principal ideal of R contained in f_αR is generated by a central idempotent element; (3) every set of central orthogonally idempotent elements of R contained in f_αR is finite one.2. R≌multiply from i∈1 R_i, where R_i is a Artinean simple ring, iff there exists a set {f_α}α∈J of central idempotent elements of R satisfying (1) (as 1, (1)) ; (2) every principal right ideal of R contained in f_αR is generated by a idempotent element; (3) every set of orthogohally idempotent elements of R contained in f_αR is finite.