The finitely generated-ness of ideals $I +rR$ and $I:r$ imply $I$ is a finitely generated ideal

Question

Let $I$ be an ideal of a commutative ring $R$, and let $r ∈ R$. Show that if the ideals $I +rR$ and $I:r=\{s∈R:sr∈I\}$ are finitely generated, then $I$ is a finitely generated ideal.

Can anyone give some help or hints, thanks a lot.

Answer 1

Suppose that $(I,r)=(a_1+rx_1,\ldots,a_k+rx_k)$ and $(I:r)=(b_1,\ldots,b_s)$, with $a_i\in I$ and consider $a_1,\ldots,a_k,rb_1,\ldots,rb_s$.

Spoiler

Note that if $x\in I$ then we can write $x=\sum r_ia_i+r \sum r_i x_i$. Then $\sum r_ix_i\in (I:r)$, say it equals $\sum s_ib_i$, and hence $x=\sum r_ia_i+\sum s_irb_i$.

Answer 2

Hint. $0\to r(I:r)\to I\to (I+rR)/rR\to 0$ is a short exact sequence of $R$-modules.