0

An Exercises from Linear Algebra by Serge Lang

Let R be a commutative ring. If $M$ is an ideal, denote the ideal $MM$ by $M^2$. Let $M_1, M_2$ be two ideals such that $M_1+M_2=R$. Show that $M_1^2+M_2^2=R$.

I let $x \in M_1,y \in M_2 $ such that $x+y=e$ (the identity element). Then I want to prove there exist some elements $a \in M_1^2, \ b \in M_2^2$ respectively and such that $a+b=e$. However I am stuck here. any hint?

user26857
  • 52,094
Math The Novice
  • 908

1 Answers1

3

Fill in details:

$$R=R^3=(M_1+M_2)^3\subset M_1^2+M_2^2$$

Further hint: Using the binomial theorem( why can we?) show that every element in $\;(M_1+M_2)^3\;$ is either in $\;M_1^2\;$ or in $\;M_2^2\;$.

DonAntonio
  • 211,718
  • 17
  • 136
  • 287