Prove that this is an ideal of a commutative ring

[annoymage]

Homework Statement

Let R be a commutative ring, $c \in R$, M is ideal in R

prove that $J=\left\{rm+c\ |\ m \in M \ and\ r \in R \right\}$ is ideal in R

Homework Equations

n/a

The Attempt at a Solution

for non-emptiness is easy

so i want to show any x=rm+c, y=r'm'+c

x-y=rm-r'm'-c+c is in J clue please T_T

 

[HallsofIvy]

x- y= r m- r' m'- c+ c= r m- r'm'. Now use the fact that m is in M which you are told is an ideal.

 

[annoymage]

rm-r'm' is in M right? then T_T more clue