Some Results on Semilocal Rings

We first give simple proofs to some main results on semilocal rings of Zhang (Proc. Amer. Math. Soc. 137, 845-852, 2009). Then, as an extension, we prove that the R/J (R)-module (aR) over bar is semisimple if and only if every descending chain in aR : a(1) R. superset of center dot center dot center dot superset of a(n)R superset of center dot center dot center dot with a(i)+1 = a(i) - a(i)b(i)a(i) (i = 1, 2,...) eventually terminates.