In this note we shall investigate a topological version of the problem of Kurosh: “Is any algebraic algebra locally finite?” Kaplansky’s theorem concerning the local nilpotence of nil PI-algebras is well-known. We will prove a generalization of Kaplansky’s theorem to the class of locally compact rings. We use in the proof a theorem of A. I. Shirshov [8] concerning the height of a finitely gener...

A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [3] completely characterized the commutative local rings R for which Mn(R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring Mn(R) is strongly clean. Here we completely determi...

A ring R is said to be n-clean if every element can be written as a sum of an idempotent and n units. The class of these rings contains clean ring and n-good rings in which each element is a sum of n units. In this paper, we show that for any ring R, the endomorphism ring of a free R-module of rank at least 2 is 2-clean and that the ring B(R) of all ω × ω row and column-finite matrices over any...

An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently, Anderson and Camillo (Anderson, D. D., Camillo, V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as we...

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...

A ring R is called clean if every element of the sum a unit and an idempotent. Motivated by question proposed Lam on cleanness von Neumann Algebras, Vaš introduced more natural concept for ?-rings, ?-cleanness. More precisely, ?-ring ?- projection (?-invariant idempotent). Let F be finite field G abelian group. In this paper, we introduce two classes involutions group rings form characterize ?-...

Let R be a commutative local ring. It is proved that R is Henselian if and only if each R-algebra which is a direct limit of module finite R-algebras is strongly clean. So, the matrix ring Mn(R) is strongly clean for each integer n > 0 if R is Henselian and we show that the converse holds if either the residue class field of R is algebraically closed or R is an integrally closed domain or R is ...