Let a monoid S act on a ring R by injective endomorphisms. An over-ring A(R; S) of R is called the S-Cohn-Jordan extension of R if (1) the action of S on R extends to an action of S on A(R; S) by automorphisms and (2) for any a ∈ A(R; S), there exists s ∈ S such that s⋅a∈ R.

A classical result of P.M. Cohn, which was originally formulated in much more general context of Ω-algebras (instead of rings), says that such an extension always exists provided the monoid S possesses a group S^-1S=G of left quotients.

The aim of the talk is to present a series of results relating various algebraic properties of R and that of A(R; S). For example primeness, Goldie conditions and other finiteness conditions will be considered.

Some possible applications to the skew semigroup rings R # S and skew polynomial rings R[x;σ,δ] will be also discussed.