Prelim Problem 26+27 J.O

26. (Preliminary for T, April 3; finalized Sat, April 14) a. Prove that equality modulo an ideal I of a ring R is an equivalence relation. (In our class, you may always assume (unless stated otherwise) that a ring is commutative and has 1.)

To be an equivalence relation, must have reflexivity, symettricity, and transitivity.
Suppose we have two elements $a,b\in R$
Reflexivity: $a\sim_{I} a$

Symetricity: If $a\sim_{I}b,$ then $b\sim_{I}a$.
we have a~b. a+(-a) ~ b+(-a)
0+(-b)~b+(-a)+(-b)
2b+(-b)~2a+(-a)
b~a

Transitivity: If a~b and b~c, then a~c

Since a~b and b~c, then (a-b) and (b-c) are both in the ideal

In order for a~c to work, then a-c must be in the ideal
By adding (a-b) to (b-c) we get (a-c) which shows that a~c

b. Prove that multiplication on the quotient ring R/I is well defined: (r+I)(s+I)=rs+I.

If $(r+I)+(s+I) = (r+s)+I$, this is only an operation on the first two items in the parentheses.
So $(r+_{R}I)\cdot_{R/I}(s+_{R}I)=(r\cdot_{R}s+_{R}I)$
$\Longrightarrow (r+I)(s+I)=rs+I$

(Preliminary for T, April 3; finalized Sat, April 14) a. Show that every unit of a ring R has a unique inverse.
Suppose there is $a,b$ such that $ab=1$.
Suppose there is then another element $c$ such that $ac=1$.
Then $b=c$.
b. Show that the units of Z/nZ are precisely those [a] such that GCD(a,n)=1.
This means that a,n are relatively prime (or co-prime).
The units of intergers mod n are all elements a such that their GCD is 1.
The elements must both be relatively prime because if they were not, then there would be units outside of the $\mathbb{Z}/n$ depending on any $n\in \mathbb{N}$. This would be troublesome. And so their GCD would have to be 1.

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