Condition for cyclic module

Question

Let A, B be cyclic R-modules. Is there any condition of R to ensure that $M=A\oplus B$ is a cyclic R-module?

Answer 1

For commutative rings, this is impossible. In other words, if $R$ is any commutative ring, there exist two cyclic $R$-modules $A,B$ such that $A \oplus B$ is not cyclic.

proof:

Let $M$ be any maximal ideal of $R$. Then $R/M=k$ is a field, and it is a cyclic $R$-module. Let $$A=B=k$$ Suppose that $A \oplus B$ is cyclic as an $R$-module. Then, it has a generator, say $x$.

Then $x$ generates $A \oplus B$ also as a $k$-module. This means that $$\dim_k (A \oplus B)=1$$ But this is a contradiction, since $$A \oplus B = k \oplus k = k^2$$ which has dimension $2$.