Rings in which elements are a sum of a central and a unit element

In this paper we introduce a new class of rings whose elements are a sum of a central and a unit element, namely a ring RR is called CUCU if each element a∈Ra∈R has a decomposition a=c+ua=c+u where cc is central and uu is unit. One of the main results in this paper is that if FF is a field which is not isomorphic to Z2Z2, then M2(F)M2(F) is a CUCU ring. This implies, in particular, that any square matrix over a field which is not isomorphic to Z2Z2 is the sum of a central matrix and a unit matrix.