Is $R_1 \oplus 0$ a free $R_1 \oplus R_2$-module?

Question

Suppose $R_1$ and $R_2$ are unital rings. Consider $R_1 \oplus \{0\}$ an $R_1 \oplus R_2$-module. Is this a free module?

I am thinking it's not, since there are relations. How can I take this idea further?

Answer 1

Certainly not, since it has a non-zero annihilator: $\{0\}\times R_2$.

Answer 2

Your example is actually an ideal of a ring: in the commutative case it is a free module iff it is principal and the generating element is not a zero-divisor. In your case it is principal being generated by $(1,0)$ but is n a zero divisor as pointed out by Bernard, and hence not free.