(related to Lemma: Factor Rings, Generalization of Congruence Classes)
We will show the lemma in five steps, by proving the existence of an additive group \((R/I,\oplus)\), introducing a multiplication operation "\(\odot\)", and proving the ring properties of the resulting algebraic structure \((R/I,\oplus,\odot)\).
By hypothesis, \(I\lhd R\) is an ideal of the ring \( ( R, + ,\cdot )\). Thus, by definition of any ideal, \( ( I, + )\) is a subgroup of the additive group \((R, + )\). Moreover, since \( ( R, + ,\cdot )\) is a ring, the group \((R,+)\) and its subgroup \((I , + )\) are commutative. Therefore, \((I , + )\) is a normal subgroup of \((R , + )\). According to the corresponding lemma, there exists an (here additive) factor group \((R/I ,\oplus)\), while the addition "\(\oplus\)" is defined by
\[(r_1 + I)\oplus(r_2+I):=(r_1 + r_2)+I\quad\quad(r_1,r_2\in R).\]
According to this lemma, the operation "\(\oplus\)" is well-defined, because it does not depend on the particular choice of representatives \(r_1,r_2\). Also, the elements of \((R/I ,\oplus)\) are residue classes \(r+I~(r\in R)\), its identity element is \(I\). The inverse of an element \((r + I)\in R/I \) is \( (-r + I)\in R/I \).
This follows immediately from the commutativity of the group \((R, + )\) and the definition
\[(r_1 + I)\oplus(r_2+I)=(r_1 + r_2)+I=(r_2 + r_1)+I=(r_2 + I)\oplus(r_1+I).\]
To realize it, suppose \(r_1,s_1\in R\) are two different representatives of the residue class \(r_1 + I\) in \(R/I\) and \(r_2,s_2\in R\) are two different representatives of the residue class \(r_2 + I\) in \(R/I\). We have to show that \[(r_1 \cdot r_2)+I=(s_1 \cdot s_2)+I.\] By our assumption \[\begin{array}{ccl} s_1=r_1+i_1\quad (i_1\in I),\\ s_2=r_2+i_2\quad (i_2\in I).\\ \end{array}\] Therefore \[(s_1\cdot s_2) + I=((r_1+i_1)\cdot(r_2+i_2))+ I=(r_1\cdot r_2+\underbrace{r_1\cdot i_2+i_1\cdot r_2+i_1\cdot i_2}_{\in I,\text{ since }I\text{ is an ideal }})+ I=(r_1\cdot r_2)+ I.\]
Thus, the multiplication defined like this does not depend on the representatives \(r_1,r_2\in R\).
The associativity follows for any \((r_1 + I),(r_2 + I),(r_3 + I)\in R/I\) from
\[\begin{array}{ccl} (r_1+I)\odot((r_2+I)\odot(r_3+I))&=&(r_1+I)\odot((r_2\cdot r_3)+I)\\ &=&(r_1\cdot(r_2\cdot r_3))+I\\ &=&((r_1\cdot r_2)\cdot r_3)+I\quad\quad("\cdot"\text{ is associative!})\\ &=&((r_1\cdot r_2)+I)\odot (r_3+I)\\ &=&((r_1+I)\odot(r_2+I))\odot (r_3+I)\\ \end{array}\]
Note, that if \((R,\cdot)\) is a monoid, so is \((R/I,\odot)\). For if \(1\) is the identity element of \((R,\cdot)\), then the identity element of \((R/I,\odot)\) is \((1 + I)\), since for any \(r+I\in R/I\)
\[\begin{array}{ccl} (r+I)\odot(1+I) = (r\cdot 1)+I=r+I,\\ (1+I)\odot(r+I) = (1\cdot r)+I=r+I.\\ \end{array}\]
Therefore, \(R/I\) is an unit ring, if and only if \(R\) is.
For any \((r_1 + I),(r_2 + I),(r_3 + I)\in R/I\) we have:
\[\begin{array}{ccl} (r_1 + I)\odot((r_2 + I)\oplus(r_3 + I)) &=&(r_1 + I)\odot((r_2 + r_3) + I)\\ &=&(r_1 \cdot(r_2 + r_3)) + I\\ &=&((r_1 \cdot r_2) + (r_1 \cdot r_3)) + I\\ &=&((r_1 \cdot r_2)+I)\oplus((r_1 \cdot r_3) + I)\\ &=&((r_1+I) \odot (r_2+I))\oplus((r_1+I) \odot (r_3 + I)) \end{array}\] and \[\begin{array}{ccl} ((r_1 + I)\oplus(r_2 + I))\odot(r_3 + I) &=&((r_1 + r_2)+ I)\odot(r_3 + I)\\ &=&((r_1+r_2)\cdot r_3) + I\\ &=&((r_1 \cdot r_3) + (r_2 \cdot r_3)) + I\\ &=&((r_1 \cdot r_3)+I)\oplus((r_2 \cdot r_3) + I)\\ &=&((r_1+I) \odot (r_3+I))\oplus((r_2+I) \odot (r_3 + I)). \end{array}\]
We have proven the resulting algebraic structure \((R/I,\oplus,\odot)\) to be a ring, es required.
