Operations with Ideals

notering-theory

Given ideals $I,J⊴(R,+,\cdot )$, the following are also ideals.

1. $I\cap J$.
2. $I+J=\{x+y\mid x\in I,y\in J\}$.
3. $I\cdot J=\{x_1{y}_1+x_2{y}_2+\dots +x_n{y}_n\mid n\ge 1,x_i\in I,y_i\in J\}$.

Proofs:

1. …
2. …
3. If $x_1,\dots ,x_n\in I,y_1,\dots ,y_n\in J$, $x_1{y}_1+\dots +x_n{y}_n\in IJ$. If $r\in R⟹r\cdot (x_1{y}_1+\dots +x_n{y}_n)=(r{x}_1)y_1+\dots +(r{x}_n)y_n$. If $x_{n+1},\dots ,x_{n+m}\in I,y_{n+1},\dots ,y_{n+m}\in J,x_{n+1}{y}_{n+1}+\dots +x_{n+m}{y}_{n+m}\in IJ$…