Remarks on the Jacobson radical

Source abstract

The aim of the paper is to investigate the set Δ ( R ) =: { r ∈ R ∣ r + U ( R ) ⊆ U ( R ) } \Delta (R) =: \{r\in R\mid r+U(R)\subseteq U(R)\} of a ring R R . This set is a ring closely related to the Jacobson radical of R R . It is shown that Δ ( R ) \Delta (R) is the largest Jacobson radical subring of R R which is closed with respect to multiplication by units of R R . The behavior of Δ \Delta under ring constructions is studied, some families of rings for which Δ ( R ) = J ( R ) \Delta (R)=J(R) are presented. Methods of constructing rings with Δ ( R ) ≠ J ( R ) \Delta (R)\ne J(R) are also described.