Two classes of rings generated by their units

Stable Rings Generated by Their Units

We introduce the class of rings satisfying (m,1)-stable range and investigate equivalent characterizations of such rings. These give generalizations of the corresponding results by Badawi (1994), Ehrlich (1976), and Fisher and Snider (1976). 2000 Mathematics Subject Classification. 19B10, 16E50. Let R be an associative ring with identity. A ring R is said to have stable range one provided that ...

Groups generated by two bicyclic units in integral group rings

Some classes of strongly clean rings

A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

ON NEW CLASSES OF MULTICONE GRAPHS DETERMINED BY THEIR SPECTRUMS

A multicone graph is defined to be join of a clique and a regular graph. A graph $ G $ is cospectral with graph $ H $ if their adjacency matrices have the same eigenvalues. A graph $ G $ is said to be determined by its spectrum or DS for short, if for any graph $ H $ with $ Spec(G)=Spec(H)$, we conclude that $ G $ is isomorphic to $ H $. In this paper, we present new classes of multicone graphs...

A pr 2 00 9 Group Rings that are Additively Generated by Idempotents and Units

Let R be an Abelian exchange ring. We prove the following results: 1. RZ2 and RS3 are clean rings. 2. If G is a group of prime order p and p is in the Jacobson radical of R, then RG is clean. 3. If identity in R is a sum of two units and G is a locally finite group, then every element in RG is a sum of two units. 4. For any locally finite group G, RG has stable range one. All rings in this note...