Prove the equivalence of the standard definition of equivalence relation on a group and the given alternative definition (of equivalence relation on a group).

Show that the following are equivalent: (a) ~ is an equivalence relation on a group G (b) ~ is reflexive and, for all elements a, b, c of G: if a ~ b and b ~ c, then c ~ a.

Show that a+b is a unit of a commutative ring with identity.

Let R be a commutative ring with identity. Let a,b in R. Suppose that a is a unit, and b^2 = 0. Show that a+b is a unit. See attached file for full problem description.

Properties of Elements of a Ring

Give an example of two elements a,b in a ring R such that a(b)=0 but b(a) <> 0. See attached file for full problem description. keywords: property

Annihilators and Ideals

Let R be a commutative ring and let A be any subset of R. The annihilator of A, denoted by Ann(A), is the set {r in R:r(a)=0 for all a in A}. Show that Ann(A) is an ideal of R. See attached file for full problem description.

Continuous Real Functions and Subrings

Let R be the ring of continuous functions from the reals to the reals. Define A={f in R: f(0) is an even integer}. Show that A is a subring of R, but not an ideal. See attached file for full problem description.

Prove that a subgroup is expressible as the union of conjugacy classes if and only if it is a normal subgroup.

Let G be a group, and let H be a subgroup of G. Prove that H is a normal subgroup if and only if H can be expressed as the union of conjugacy classes of G.

Ring Theory

If R is a ring and p(x) is included in R[x] then f(x) is the associated polynomial function from R to R. Find a p(x) included in Zmod2[x] such that f(x)=0 for all x included in zmod2. I know that Zmod2 is all the polynomials whose coefficients are 0 and 1 but I have no idea what I am I trying to look for.

Subgroups and Subsets

Let K and H be subgroups of G. Prove that If H union K is a subgroup of G then either H is a subset of K or K is a subset of H.

Ring theory question about polynomial rings

Let R={f(x) included in Q[x]: the coefficient of x in f(x) is 0} Prove that R is a subring of Q[x].

Areas and Dimensions of Rectangles

Peter and Ray built a bike ramp from 20 sheets of 4ft. X 8ft. plywood and propped it up on one end with some bricks. The finished ramp is rectangular. Show 3 ways they might have arranged the wood and give the measurment for each side. Assuming that they used every sheet of plywood, what is the total area of the ramp regardless ...continues