An amalgamated duplication of a ring along an ideal: The basic properties

We introduce a new general construction, denoted by R (sic) E, called the amalgamated duplication of a ring R along an R-module E, that we assume to be an ideal in some overring of R. (Note that, when E-2 = 0, R (sic) E coincides with the Nagata's idealization R(sic) E.) After discussing the main properties of the amalgamated duplication R (sic) E in relation with pullback-type constructions, we restrict our investigation to the study of R (sic) E when E is an ideal of R. Special attention is devoted to the ideal-theoretic properties of R (sic) E and to the topological structure of its prime spectrum.