Abstract Algebra Answers

Let R be a commutative Ring with identity.let I and J be ideals of R such that I+J=Show that IJ= intersection of I and J.

Give two distinct maximal ideals in the polynomial ring Q[x] with justification.

Prove that R⁵/R² is isomorphic to R³.

Give an example of infinite ring of characteristic 7.

Show that the ideal <x^2+1> is not prime in Z2[x]

Assume that (G,*) is a group and that (H,*) and (K,*) are subgroups of (G,*).Prove that (H intersects K , *) is a subgroup of (G,*)

Prove that if (G,*) is a finite group, and g is an element of G, then there exists a positive integer n such that g^n =e

Show that d:Q[x]\{0}→NU{0}:d(f)=2^deg f is a Euclidean valuation on Q[x].

Find the quotient field of integral domain {a+ib such that a,b belongs to Z}

If G is a group of even order, prove that it has an element 'a' which is not equal to 'e' satisfying a^2=e. e is identity element.