Algebraic Properties of Semi-Direct Sums of Rings

Let R be an associative ring not necessarily with unity. We say that R is a semi-direct sum of rings S and I, if R = S + I, where S is a subring of a ring R, I is an ideal of R and S ∩ I = {0}.

The aim of this paper is to investigate certain algebraic properties of semidirect sums of associative rings with applications to amalgamated rings. We generalize several results from the literature to associative rings without unity. In particular we show that the class of semi-direct sums of rings is equal to the class of amalgamated rings, we provide a description of the Jacobson radical of semi-direct sums and we offer a characterization of semi-direct sums that are left Steinitz rings.