The Structure Of Ring R[D, C] And Its Characterizations

Let D be a ring, C be a subring of D and 1_D∈ C. Write:S = R[AC] ={(d₁,… ,d_n,c,c,…)|d_i ∈D,c∈C,n≥1}, S' = R{D,C} = {(d₁…,d_n,c_n+1 ,c_n+2…)|d_i ∈D,c_j∈C,n≥1}.With addition and multiplication defined componentwise, both R[D, C] and R{D, C} are rings.It's clear that R[D,C] is a subring of (?)D. R[D,C] plays an important role in constructingnew classes of rings, getting counter example in ring theory and characterizing the relationships between different rings. Thus, it's necessary to study the structure , properties of R[D,C] and its conditions to be different classes of rings.In this paper, we first give the structure of R[D,C], including its maximal, minimal right ideals and the structure of its Jacobson radical, socle and singular right ideal as well. Then we list the classes of rings which R[D, C] can not be. At last , various conditions are obtained for R[D, C] to be (m, n)-coherent, pseudo-coherent, n-P-injective, clean, regular, rings with stable range I, internal cancellation and some other classes of rings, respectively.