I am having a very hard understanding quotient rings. I never really understood quotient groups in abstract 1 but did well enough to get by. Now I am in abstract 2 and we hit quotient rings. I really want to understand quotient rings this time around.
We don't use a specific textbook in my class just handouts...but these rings may pr may not be commutative but with unity. From my understanding quotient rings are with $R$ being a ring and $I$ an ideal of $R$ then $R/I$ is a ring with the additive operation such that $(a+I)+(b+I)= (a+b)+I$. Likewise multiplication is defined as $(a+I)(b+I)= (ab+I)$. (Where $a$ and $b$ are elements of $R$)
I am having trouble understanding how to represent the quotient ring $R/I$. Is every element (which is a coset) in the form of an $r+I$?
Can someone help me go through what should be trivial (but not to me )of why $R/R = 0$ (zero ring) and $R/0 = R$.
I think that $R/R$ is in the form of $(r+R)$ but I don't see what to do to get zero.
$R/0$ is going to be in the form of $r+0=r$. Do I now say for all $r$ in $R$ ?
I understand these proofs should be very trivial and easy but I feel like going through in detail of why exactly this happens will help me solidify what I know about quotient rings so I can actually do my homework (which are more involved)