J-semicommutative环的性质

环冗称为J—semicommutative若对任意B，b∈R由ab=0可以推得aRb∈J（R），这里J（R）是环R的Jacobson根．环R是J—semicommutative环当且仅当它的平凡扩张是J—semicommutative环当且仅当它的Don＇oh扩张是J—semicommutative环当且仅当它的Nagata扩张是，一semicommutative环当且仅当它的幂级数环是J—semicommutative环．若R／J（R）是semicommutative环，则可得到R是J-semicommutative环．本文进一步论证了如果，是环月的一个幂零理想，且R／I是J—semicommutative环，则R也是J-semicommutative环最后给出了J—semicommutative环与其他一些常见环的联系

Some Results on J-semicommutative Rings

A ring R is called J-semicommutative ring if a,b ∈ R with ab=0 implies aRb ∈ J(R) where J(R) de-notes the Jacobson radical of R.It is proved that a ring is J-semicommutative if and only if the trivial extension is J-semicommutative if and only if Dorroh extension or Nagata extension is J-semicommutative if and only if so is the power series ring of R.It is showed that if R/J(R) is a semicommutative ring,then R is a J-semicommutative ring.We also get the results that if I is a nilpotent ideal of a ring R,R/I is J-semicommutative,then R is J-semicommutative.Lastly we give the relationship between J-semicommutative rings and other related rings.