All rings are assumed to be associative and to have non-zero identity elements. Throughout \(A\) is a ring with Jacobson radical \(J(A)\), \(B\) is a unitary subring in \(A\), \(\{A_i\}_{i=1}^\infty\) is a countable set of copies of \(A\), \(D\) is the direct product of all the rings \(A_i\), \(B'\) is the subring \(\{(b,b,\dots)\mid b\in B\}\) of \(D\) and \(R(A,B)\) is the subring in \(D\) generated by the ideal \(\bigoplus_{i=1}^\infty A_i\) and the subring \(B'\). A ring \(X\) is called: (i) an \(I_0\)-ring if every right ideal of \(X\) that is not contained in \(J(X)\) contains a non-zero idempotent; (ii) an exchange ring if for any element \(a\in X\), there exists an idempotent \(e\in aX\) with \(1-e\in(1-a)X\); (iii) a right max ring if every non-zero right \(A\)-module has a maximal submodule. The author proves that the ring \(R(A,B)\) is semiprimitive (semiprime, reduced, \(I_0\)-ring) if and only if \(A\) has the same property. Also, the ring \(R\) is shown to be regular (\(\pi\)-regular, strongly \(\pi\)-regular, exchange ring, right max ring) if and only if \(A\) and \(B\) have the same property.