A commutative ring is called a local ring if it has a unique maximal ideal. Prove that if R is a local ring with maximal ideal M then every element of R-M is a unit. Prove conversely that if R is commutative ring with one in which the set of nonunits forms an ideal M, then R is a local ring with unique maximal ideal M.
1. Prove that gcd (a, lcm[b, c]) = lcm[gcd(a,b), gcd(a,c)]. 2. Find the simultaneous solutions of the following congruences: 2x ≡ 1(mod 5) 3x ≡ 9 (mod 6) 4x ≡ 1 (mod 7) 5x ≡ 9 (mod 11)
Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is fn, where {fn} is the Fibonacci sequence defined
Indicate whether each system is independent, inconsistent or dependent 4 ( 1-x) + y = 3 3 (1-y) - 4x = -=4y 1/3x - 1/6y = 1/3 1/6x + 1/4y= 0
Indicate whether each system is independent, inconsistent or independent x-5y= 4 4x + 8y = -5
Factor 4z^2 + 16z + 15.
Factor 25-9z^2
See attached file for full problem description. 1) An open-top box is to be constructed from a 4 by 6 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out. a) Find the function V that represents the volume of th
Solve for x. Your answer should be in the simplest form fraction. 3x + 2/3 = 1/6x - 3/2
Solve for x: -6 (x - 9) = 7x + 1 - 4 (-8x + 8) Please try to get the answer in the simplest form possible.
Write each expression in the form a + bi, where a and b are real numbers. (3-5i) - (2-7i)
Rational Exponents. -25^(-3/2)
Find the root. See attached file for full problem description.
Solve for R. (distance, rate, & time). See attached file for full problem description.
b/3=-3/4
Equation w/ two solutions. (a+4)/2 = (a+4)/a
Soving Equation: X+2-2 square root X+2=0
The line through (-4,6) that is parallel to y=5x The line through (3,5) that is parallel to the x-axis
10 34 ___ = _______ x x + 12
Factoring Quadratic Equations How do you factor z^2 - z - 30?
A/y + 1/3 = B/y
1/a + 1/b = 1/f for a P1V1 P2V2 ______ = ______ for P2 (note: the 2 is a tiny #2 below the P) T1 T2
Three problems 5/x + 6/x =12 a-1/a^2-4 = 1/a-2 = a+4/a+2 z/z+1 - 1/z+2 = 2z+5/z^2 + 3z +2
Please address the following question: Convert each rational expression into an equivalent rational expression with the LCD as the denominator. y x ___ , _____ 4x 6y
How do you do these type of problems? 2.4=3(m+4) 5(1-2w)+8w=15 If 3(x+1)=7 what is the value of 3x
Please complete the following: Solve the following equation for Z. (4z - 5)/2 - (z + 9)/4 = 4 Simplify the answer as much as possible.
The final velocity, V, of an object is given by the equation - See attached file for full problem description.
The centripetal force F acting on a mass M traveling along a circular path of radius R at a velocity V is given by...See attached file for full problem description.
Solve the following equation for X. Write your answer as a fraction in simplest form. 7(x-3)=-3x+2-3(-2x+7)
Customers of a phone company can choose between two service plans for long distance calls. The first plan has a $29 one-time activation fee and charges $0.11 cents a minute. The second plan has no activation fee and charges $0.15 cents a minute. After how many minutes of long distance calls will the costs of the two plans be equ