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

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?

## 2 Answers

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

• Thanks! I actually just found out that result. I am pretty new to module theory; is there a way to show nontrivial annihilator $\implies$ not free using only the definition of a free module as one with a basis? – TorsionSquid May 5 '15 at 0:20
• Here it is simple: as it's a cyclic module; i. e. a quotient of $R_1\times R_2$ by a non-zero ideal, generated by $\overline{(1,0)}$, which is killed by any $(0,r_2)$. – Bernard May 5 '15 at 0:34
• Unless $R_2 = 0$. :) – darij grinberg May 5 '15 at 3:00
• Yes. Impilcitly, I consider rings with $1$. – Bernard May 5 '15 at 7:26
• @Bernard $\{0\}$ is a ring with identity! ;-). By the way, over rings without an identity, the notion of free module doesn't make much sense: even the ring itself has no basis. – egreg May 5 '15 at 13:21

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.