Abstract Algebra Answers

Let R and S be rings. The direct product of R and S,
R ⊕ S = {(r, s) : r ∈ R, s ∈ S}
is a ring, where
(r1, s1) + (r2, s2) = (r1 + r2, s1 + s2)
(r1, s1) x (r2, s2) = (r1 x r2, s1 x s2)

(a) List the elements of Z2 ⊕ Z3 and Z3 ⊕ Z2.
(b) Are they structurally the same (that is isomorphic)? If so, how should an
element in Z2 ⊕ Z3 be identified in Z3 ⊕ Z2?

Consider the function f : Z4 → Z12 defined by f(x) = 3x. Show that f(x) does
NOT preserve multiplication.

Prove that 5Z is both a prime and maximal ideal of Z

Show that sin b

find two straight lines tangent to y=x^3 and they pass through point (2,8)

a 180 pound person burns 284 calories playing basketball for 25 minutes.at this rate,how many calories would the person burn in 45 minutes to the nearest whole number

A wall of a house is in th shape of trapezoid. Find th number of square feet of area. If a one-gallon can o paint covers approximately 200ft sq, how many cans must be purchased?
20ft, 22ft, and 8ft.

Find each of the following for the circle.
Radius
Diameter
Circumference formula
Circumference of the circle

Through this chapter,you will be using and comparing fractions and decimals.Cicle all of the fractions below that are equal to 1/2.
6/12 15/30 4/7 5/10 1/3 2/3 9/12 7/14