关于亚直既约环为体的条件

CONDITIONS THAT A SUBDIRECTLY IRREDUCIBLE RING IS A DIVISION RING

证明了亚直既约环R为体的充要条件是R的心不含非零幂零元,且R的含于心内的主左理想具有降链条件,得到了交换的亚直既约环为域的充要条件为R的心是幂等的,此外还证明了心为强正则的亚直既约环为体。

We prove that a subdirectly irreducible ring R is a division ring if and only if the heart H of R has no any nonzero nilpotent element, and the principal left ideals of R contained in H satisfy the descending chain condition. We obtain that a subdirectly irreducible commutative ring R is a field if and only if the heart H of R is idempotent, or if and only if H does not contain a nonzero nilpotent element, or if and only if H does not contain a right or left zero divisor in R. We also prove that if the heart H of subdirectly irreducible ring R is strongly regular then R is a division ring.