Instructor: Jeremy Johnson

Due Wed. Jan. 12 in class

- for all a, b in R, a+b is in R.
- for all a, b, c in R, (a+b)+c = a+(b+c)
- there exists an element 0 in R such that for all a in R a+0 = 0+a = a.
- for all a in R there exists an element -a in R such that a + -a = 0 and -a + a = 0.
- for all a, b in R, a+b = b+a
- for all a, b in R, a*b is in R.
- for all a, b, c in R, (a*b)*c = a*(b*c)
- there exists an element 1 in R such that for all a in R a*1 = 1*a = a.
- for all a, b, c in R, (a+b)*c = a*c + b*c and c*(a+b) = c*a + c*b.

- if a is nonzero and a*b = a*c then b = c.

- for all a,b in I, a+b is in I.
- for all a in I and r in R, a*r and r*a are in I.

- c|a and c|b
- if d|a and d|b then d|c

- for all a and b in Z, there exist integers q and r such that a = b*q + r, with 0 <= |r| < |b| [r = rem(a,b)]

- gcd(a,b) = gcd(b,r), where r = rem(a,b)
- gcd(a,0) = a

- if b = 0 then return a.
- return IGCD(b,rem(a,b)).

- a1 = q1*a2 + a3, 0 <= a3 < a2
- a2 = q2*a3 + a4, 0 <= a4 < a3
- ...
- a_{n-1} = q_{n-1}*a_n + a_{n-1}, 0 <= a_{n-1} < a_n
- a_n = q_n*a_{n+1}

- Let R be a ring and let I be an ideal in R. Define a equivalent to
b mod I iff a - b is in I (a equivalent to b mod n in the integers is
a special case of this). Show that this defines an equivalence relation.
Recall that an equivalence relation must satisfy the following
three properties:
- a is equivalent to a (reflexivity)
- if a is equivalent to b, then b is equivalent to a (symmetry)
- if a is equivalent to b and b is equivalent to c then a is equivalent to c (transitivity)

- The elements of the quotient ring R/I are the sets I + r, where r is in R (r is called the representative of the set I + r). Note that the set I + r can be represented in many different ways. If r' = r + i for some i in I, then I + r = I + r'. Addition and multiplication in R/I are defined by (I + r1) + (I + r2) = (I + r1 + r2) and (I + r1)*(I + r2) = (I + r1*r2). Show that these operations are well defined. That is that they are independent of the chosen representatives for I + r1 and I + r2. Also show that R/I satisfies the properties of a ring.
- Use the Euclidean remainder sequences to prove that if gcd(a,b) = 1 and a|bc then a|c.
- Use the result of the previous question to show that Z/p (integers mod p, where p is a prime) is an integral domain. Hint: show that there are no zero divisors.
- Show that there are no non-trival ideals in Z/p (i.e. the only ideals are (0) and Z/p itself. Hint, show that for any a, not equal to zero, in Z/p, that (a), the ideal generated by a is equal to Z/p.