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 are associative with identity. An element of a ring is said to be clean if it is a sum of a unit and an idempotent. A ring R is said to be clean if its every element is clean. These rings were introduced by Nicholson in [N1] as a class of examples of exchange rings. In [N1, Proposition 1.8] Nicholson proved that an Abelian exchange ring is clean. This work is motivated by the paper [M] of McGovern where it is proved that for a commutative clean ring R, the group ring RZ2 is clean. We extend this result by proving that RZ2 is clean whenever R is an Abelian exchange ring. Moreover our proof is quite short. We also prove that RS3 is clean for any Abelian exchange ring R. Let R be a commutative clean ring and G be a finite group of prime order p such that p is invertible in R. In [HN, Example 1], Han and Nicholson gave an example to show that the group ring RG may not be clean. We prove that if R is an Abelian exchange ring and G is a group of prime order p such that p ∈ J(R), then RG is clean. A lot of people have studied rings in which every element is a sum of two units (see [KS] and its references). An obvious necessary condition for the identity element of a ring R to be a sum of two units is that R does not have a factor ring isomorphic to Z2. In [KS, Theorem] it is proved that if R is a right self-injective ring which has no factor ring isomorphic to Z2, then every element of R is a sum of two units. We prove that every element of a group A ring is said to be Abelian if its all idempotents are central