Rings

In Zn x Zm (integers modulo n and m respectively) find the characteristic of the ring.

Rings : Nilpotents

If e^2 = e, show that (1-e)re and er(1-e) are nilpotents for all r belonging to R. ALSO, if e^2 = e, show that e+(1-e)re and e+er(1-e) are idempotents for all r belonging to R

Rings: Integral Domains and Fields

Find all the roots of x^2 + 3x - 4 in Z (integers) AND Z6 (integers modulo 6) AND Z4 (integers modulo 4)

Euclidean Group : Forming a Group under Matrix Multiplication

Please see the attached file for the fully formatted problems. The Euclidean group is defined as E3 ={X R 4 4 | X = , R O3, t R 3} Where O3 is a 3 3 orthogonal matrix, therefore R is an element in O3. R 4 4 means real 4 4 matrix vector space. R 3 means real 3-dimensional vector space. 0 in ...continues

Subgrpous

Prove: any subgroup of the order of p^(n-1) in a group of order p^n, where p is a prime, is a normal subgroup

Integral domain

There is integral domain with exactly six elements. Disprove or Prove

Algebraic numbers

Is sin 1 degree an algebraic number?

Transformations in Hom(v,v)

Prove that the invertible transformations in Hom(v,v) form a group under multiplication

Automorphism

1. Let T be any automorphism of G, show that ZT<(subset) Z. If G is a group and Z is the center of G.

Isomorphism

Let T be defined on real two dimensional plain, and that: (x,y)T = (ax+by, cx+dy) ; a, b, c, d real constants. Prove that T is a vector space homomorphism. What value of a, b, c, d will T be an isomorphic or isomorphism?