Q. Pa rt 1) An element a in a commutative ring R with unity e is said to be invertible if there is an element b in R such that ab=e. Prove that if R is a commutative ring with unity, then the invertible elements form a group with respect to the multiplication of the ring.
Part 2) What is the group of invertible elements in ZZ4?
A. Write U = set of invertible elements in R.
1) Closure: a and b are in U then there exist a^-1 and b^-1 so that aa^-1 = bb^-1 = 1. Then (ab)(b^-1a^-1) = a(bb^-1)a^-1 = a(1)a^-1 = aa^-1 =1, so that ab is also in U.
2) Clearly U is associative since R is.
3) 1 is clearly in U.
4) Clearly, by the definition of U every element has an inverse.
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U(Z/4) = {1,3}. If a is invertible in Z/4 then ab = 1 mod 4 is solvable. So a can't be 2 otherwise 2a = 1 mod 4 implies 2 divides 1. If a = 1 then 1*1 = 1 mod 4, and if a = 3 then 3*3 = 9 = 1 mod 4.
A more general way to do this part is:
Using Bezout's theorem/Euclidean algorithm, gcd(a,4) = 1 implies ab = 1 +4d = 1 mod 4. The only numbers with gcd(a,4) =1 are a = 1 and 3.
What is the direct corolation of R-value to sound dampening?
Q. Owens Corning Insulating R-19 compared to R-13 would reduce/absorb x% of sound. Typically sound absorption is rated as a coefficient with 1.0 being 100% and .5 being a 50% reduction. What value would R-19 or R-13 provide in reducing sound?
A. That's going to be hard to estimate because the R value of Owens Corning insulation is resistance to HEAT/COLD and not sound or vibration.
What must be the value of r for the astronauts to weigh 1/4 of their earth weight?
Q. Suppose the surface (radius = r) of a cylindrical space station is rotating at 23.1 m/s to provide artificial gravity. What must be the value of r for the astronauts to weigh 1/4 of their earth weight?
A. Answer --> 217.6 m
Given.
v = 23.1 m/s
a = 2.4525 m/s^2 (25% of 9.81 m/s^2 Earth gravity)
Original equation is:
a = v^2 / r
Solve for r:
r = v^2 / a
With values:
r = (23.1 m/s)^2 / (2.4525 m/s^2)
r = (533.61 m^2/s^2) / (2.4525 m/s^2)
r = 217.6 m
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Title : Prove that if R is a commutative ring with unity, then the invertible elements form a group with respect to th?
Description : Q. Pa r t 1) An element a in a commutative r ing R with unity e is said to be invertible if there is an element b in R such that ab=e. Pr...