Small or Superfluous submodule

Question

Suppose $A$ is a summand of an $R$-module $M$ and take any $P \leq M=A \oplus B$ such that $P \nleq A$, $A \nleq P$, $P \nleq B$, $B \nleq P$, $(A+P) \cap B \nleq P$ and $(A+P) \cap B \nleq A$. Is it true that $(A+P) \cap B \leq_{S} A+P\ ?$

Answer 1

Here $(A+P) \cap B \nleq_{S} A+P$. If $(A+P) \cap B \leq_{S} A+P$ then $ A+P=A$ which implies $P \leq A$.