# Proof

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 $$(R/I,\oplus,\odot)$$.

### $$(1)$$ Existence of the additive group $$(R/I,\oplus)$$

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 $$(-r + I)\in R/I$$.

### $$(2)$$ the factor group $$(R/I,\oplus)$$ is Abelian

This follows immediately from the commutativity of the group $$(R, + )$$ and the definition

$(r_1 + I)\oplus(r_2+I)=(r_1 + r_2)+I=(r_2 + r_1)+I=(r_2 + I)\oplus(r_1+I).$

### $$(3)$$ In $$R/I$$, we can also define a multiplication "$$\odot$$" by $$(r_1 + I)\odot(r_2+I):=(r_1 \cdot r_2)+I$$, which does not depend on the representatives $$r_1,r_2\in R$$.

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 $$r_1,r_2\in R$$.

### $$(4)$$ The algebraic structure $$(R/I,\odot)$$ is a semigroup, i.e. the multiplication "$$\odot$$" is associative.

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.

### $$(5)$$ The distributivity laws hold for "$$\oplus$$" and "$$\odot$$".

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.

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### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013