On the Amalgamated Duplication of an Unitary Normed Algebra Along an Ideal

In this paper, based on an amalgamated duplication of a ring along an ideal, we will construct the amalgamated duplication of an unitary normed algebra X along its proper ideal I (i.e. X ?I). Then we will discus about its Banach conditions and abstract properties. Mainly we have proven that if X ba a Banach algebra and I be its closed ideal then X ?I will be Banach algebra too. As well, we have shown that its completion and ideals based on original normed algebra X. Finally some more aspects to expend these results for generalized normed algebras are given.