Math 0430 Exam #1Sample1. (40) pts. Let a, b, d, p, n  with b  0 and n > 1. Let <R, +, > and <S, , > be rings. Define or tell what is meant by the following: (a) b divides a (b| a) (b) d is the greatest common divisor of a and b (d = (a,b)) (c) p is prime (d) a is congruent to b modulo n ( a  b (mod n)) (e) u is a unit of the ring R (f ) z is a zero divisor of the ring R (g) <R, +, > is a division ring. (h) f: R  S is an isomorphism. 2. (5) State the Fundamental Theorem of Arithmetic.3. (15) State the Well-Ordering Principle or Axiom. Use it to prove one of the following: (a) Let a and b be integers with b> 0. Then there exist integers q and r such that a = qb + r and 0  r < b. (b) Let a and b be integers, not both 0 and let d be their greatest common divisor. Then there exist integers u and v such that d = au + bv and d is the smallest positive integer that can be expressed in the form au + bv. (c) Every integer n > 1 can be written as a product of primes.4. (10) Use the Euclidean Algorithm to find (30,198) and express it as a linear combination of 30 and 198.5. (10) Let a, b and c be integers. Prove that If (a, c) = 1 and (b, c) = 1, then (ab, c) = 1.6. (10) Let n = 2k where k is an odd integer. Show that k2 k(mod n). 7. (10) Let S = {00ba | a, bR }. Show that S is a subring of the ring T = {c0ba | a, b, cR } 8. (18) Let operations of “addition” and “multiplication” be defined on  by a b = a + b – 1 and ab = a + b – ab. Prove that , ,  is isomorphic to , +, , the ring of integers with ordinary addition and multiplication.9. (12) Define multiplication on Z2 Z2by (a,b)  (c,d) = (ac – bd, bc + ad). Noting that the multiplication is commutative, complete the multiplication table for Z2 Z2. Is this the multiplication table of an integral domain? Of a field? Justify your answers.10. (20) R = {a, b, c, d }, S = {r, s, t, u}, and T = {w, x, y, z} are three rings of order 4. R abcddbadcccdabbdcbaadcba dadadaccacdcbabaaaaadcba S ursttrutsssturrtsruutsru ttuutssuusrruuruuuuutsru- T wxyzzxwzyyyzwxxzyxwwzyxw yxzwzxzywyzyxwxwwwwwzyxwThe characteristic of a ring R is the least positive integer m such that ma = 0R for every aR. If no such integer exists, the ring is said to have characteristic zero. (a) Find the units in R. (b) Find the zero divisors in S.(c) Find 3-t – r-s. (d) Find y4z-1.(e) Find the characteristics of the rings. (f) Are any of the rings commutative? If so, state which.(g) Are any of the rings integral domains? If so, state which. (i) Are any of the rings fields? If so, state which. (j) Are any of the rings isomorphic to one another? If so, state which. (k) Are any of the rings isomorphic to 4? If so, state which. You may earn 5 points extra credit for each example of: (a) a ring with identity that is not commutative (b) a division ring that is not a field (c) a ring with characteristic zero that does not have an