Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every id...Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N J（R） of R, that H has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X（R） is a union of 2, 3, 4, and 5 orbits under the left regular action on X（R） by G（R）. For a semiperfect ring R which is not local, it is shown that if G（R） is a cyclic group with 2 ∈ G（R）, then R is finite. We lastly consider two sorts of conditions for G（R） to be an Abelian group.展开更多
We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that...We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.展开更多
文摘Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N J（R） of R, that H has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X（R） is a union of 2, 3, 4, and 5 orbits under the left regular action on X（R） by G（R）. For a semiperfect ring R which is not local, it is shown that if G（R） is a cyclic group with 2 ∈ G（R）, then R is finite. We lastly consider two sorts of conditions for G（R） to be an Abelian group.
基金Partially supported by the NSF (10071035) of China.
文摘We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.
