Abstract Algebra Answers

7. For each operation * given below, determine whether * is a binary operation, commutative or associative. In the event that * is not a binary operation, give justification for this.

i. On Z, a*b=a-b

ii. On Q, a*b=ab+1

iii. On Q, a*b=ab/2

iv. On Z+ , a*b= 2ab

Prove that the set Z of all integers with binary operation * defined by a*b=a+b+1

for all a, b belonging to G is an Abelian group.

cosider the ideal I={x^2-4x+3,x^3+3x^2-x-3} of the ring .find polynomial p such that I=<p>

find aba^-1 where (i)a=(5,7,9) ,b=(1,2,3)(ii) a=(1,2,5)(3,4),b =(1,4,5)

Let A be {1, 2, 3, 4}. Then the sequences 124, 421, 341 and 243 are same  permutations of A taken 3 at a time. The sequences 12, 43, 31, 24, and 21 are examples of different permutations of A taken two at a time.

a*b=-a-b-2ab; a⊕b=3a+3b

x*y=x²+2x+y²; x⊕y=x+y

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation

defined by aOb = max{ a,b}.

Give an example of a subring of a ring, say A, that is not an ideal of A

Determine which of the polynomials below is (are) irreducible over Q. a. x⁵+9x⁴+12x²+6 b. x⁴+x+1