$ I(J+L)=IJ+IL$ if $I,J,L$ are ideals of $K$ Given that $I,J,L$ are ideals of $K$, do we have $I(J+L)=IJ+IL$? 
I am confused how to do it.

$\rm\, H \supseteq I(J\!+\!L)\!\!\iff\!\! H\!:\!I \supseteq J\!+\!L \!\!\iff\!\! H\!:\!I \supseteq J,L \!\!\iff\!\! H\supseteq IJ,IL \!\!\iff\!\! H\supseteq  IJ\!+\!IL.\  $ QED
Remark $\ $ We employed the universal property of ideal sums, and ideal quotients. Recall that the ideal quotient $\rm\,H\!:\!I \,=\, \{ r\in R\ |\ rI\supseteq H\},\:$ hence $\rm\:H\!:\!I \supseteq J\!\iff\! H\supseteq IJ.$
