Is $\langle\bigcup_{i=1}^nA_i\rangle=A_1+\cdots+A_n$ always true for subsets $A_i$ in ring?

Question

Let $R$ be a ring(not necessarily commutative or having a unity). Let $A_1,~\cdots,~A_n$ be subsets of $R$. Then is $\langle\bigcup_{i=1}^nA_i\rangle=A_1+\cdots+A_n$ always true (where $\langle\rangle$ means the ideal generated by the set inside)? I can only find some authors mentioned that if $M$ is a left $R-$module, $S_1,~\cdots,~S_n$ be subsets of $M$, then $\langle\bigcup_{i=1}^nS_i\rangle=S_1+\cdots+S_n$ (where $\langle\rangle$ means the submodule generated by the set inside). But I'm afraid that it might not be true in ring theory.