In a graded ring $B=\bigoplus_{d\ge 0} B_d$, the element $0$ is homogeneous with grade $d$ for every $d\ge 0$, in fact since every $B_d$ is an additive subgroup of $B$, then it must contain $0$.
Can exist an element $b\in B\setminus\{0\}$ such that $b\in B_e\cap B_f$ with $e\neq f$ (a non-zero element with two distinct grades)? For polynomial rings for example this particular element doesn't exist.
Thanks in advance.
No. If an element $b \in B_i \cap B_j$, with $i \neq j$, then $b$ must be zero. Otherwise, the sum $B_i \oplus B_j$ is not direct.
Compare with http://en.wikipedia.org/wiki/Direct_sum_of_groups