ring theory – $ R $ – ideal algebra generated by a set

Leave $ R $ be a unital ring, and $ S subset R $. The two-sided ideal that I generated $ S $ is the set of all form elements $ displaystyle sum_ {finite} a_is_ib_i $ where $ a_i, b_i in R $ Y $ s_i in S $.

Leave $ R $ be a commutative unital ring and $ A $ be an algebra about $ R $. Leave $ S subset A $ Y $ I $ be double $ R $– ideal algebra generated by $ S $. It is true that $ I $ is the set of all form elements $ displaystyle sum_ {finite} r_ia_is_ib_i $ where $ r_i in R $, $ a_i, b_i in A $ Y $ s_i in S $?