The construction I had in mind was something like that, thanks. Still, I can't really remember why I wanted it to be an isomorphism. Thanks a lot for the answer

(c_0,...,c_n) is in S. Thus, S is a commutative unitary ring with unit 1_s=(1,0,0,0...). Moreover, all the elements (a,0,0,0...) in A are a subring A_1 of S isomorphic to A. We define x as (0,1,0,0...) and x^n as (0,0,0,0...,a,0,0,0...) with n zeroes before a. A polynomial will be an object of the form a_0+a_1 x+...+a_n_0 x^n_0. I consider this to be the "canonical definition of A[x]". I'm sorry if I wasn't clear enough. I surely will be next time.

@Dadvi: yes, n is a natural number, and I've never written about a "usual isomorphism", I asked if that could have been an alternate definition to the canonical definition of the polynomial ring over A, A[x]. Let us conisder S the ser of all the infinite sequences (a_0,a_1,...,a_n) of elements of A such that there exists an n_0 in N with the property that for all n > n_0, a_n = 0. S is a subset of A^N. The sum will be thus defined: (a_0,a_1,...a_n)+(b_0,...b_n)=(a_0+b_0,...,a_n+b_n) and the product as (a_0,...,a_n)(b_0,...,b_n)=(c_0,...,c_n), with c= Sum of (a_j *b_j) from j+k=i.

Right, what I really meant was A is a domain, but still, I guess that does not change things. Thanks for the reference