Finite rings in which commutativity is transitive.
Let \(R\) denote a ring with Jacobson radical \(J(R)\) and center \(Z(R)\). Define \(R\) to be commutative transitive (CT) if commutativity is a transitive relation on \(R\setminus\{0\}\) and weakly commutative transitive (wCT) if commutativity is transitive on \(R\setminus Z(R)\). The authors note that a CT ring cannot have a unit, but adjoining 1 yields a wCT ring; and they then proceed to study noncommutative finite wCT rings \(R\) with 1. They show that such a ring \(R\) is a direct sum of a commutative ring and an indecomposable noncommutative wCT ring of prime power order; and then they prove that if \(R_1\) is the indecomposable summand, then either (i) \(R_1=M_2(F)\) for some field \(F\), or (ii) \(R_1/J(R_1)\) is a field or a direct sum of two fields, so that \(R_1\) is local or basic. The final sections of the paper provide some information about which finite local and basic rings are wCT rings. The motivation for this paper was earlier work on CT and wCT groups. An interesting result pointed out by the authors is that for local rings \(R\), but not for general \(R\), \(R\) is a wCT ring if and only if the group of units of \(R\) is a wCT group.
scientific article; zbMATH DE number 5864132
scientific article