How can I show the following?
Show that, if $a$ and $b$ are elements of a ring $R$ and $I$ is an ideal of $R$, then $$a+I=0+I \iff a \in I$$
I am so interested to know the proof.Thanks and good night all!
How can I show the following?
Show that, if $a$ and $b$ are elements of a ring $R$ and $I$ is an ideal of $R$, then $$a+I=0+I \iff a \in I$$
I am so interested to know the proof.Thanks and good night all!
Consider the definition of $a+I$:
$$a+I=\{a+r\mid r\in I\}$$
Clearly, we have that $$0+I=\{0+r\mid r\in I\}=\{r\mid r\in I\}=I.$$
Because $I$ is an ideal, $0\in I$. Thus, for any $a\in R$, one of the elements of $a+I$ is $a+0=a$. If we knew that $a+I=0+I=I$, then $a$ is an element of $I$.
Now try showing the opposite direction on your own :)