An element a in a ring R is called nilpotent if there exists an
n such that an = 0.
(a) Find a non-zero nilpotent element in M2(Z).
(b) Let R be a ring and assume a, b ∈ R have at = 0
and bm = 0 for some positive integers t and m. Find an n
so that (a + b)n = 0. (You just need to find any n that
will work, not the smallest!)
(c) Show...
Given the sub-ring A = Z × Z ⊂ R 2 ,Where, in R
2 we have the sum operation and the complex product.
a) Prove that indeed A it is a sub-ring.
b) ¿It is A an entire domain?
c)Consider U = {(0, n) : n ∈ Z}, prove that U is an ideal of
A.
d) Calculate A/U.
e) ¿It is U an ideal of R2?
I need all the paragraphs thanks
In this exercise, we will prove the Division Algorithm for
polynomials. Let R[x] be the ring of polynomials with real
coefficients. For the purposes of this exercise, extend the
definition of degree by deg(0) = −1. The statement to be proved is:
Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there
exist unique polynomials q(x) and r(x) such that
f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)).
Fix general f (x) and g(x).
(a) Let...
10.3.6 Exercise: Product of Pairwise Comaximal Ideals. Let R be
a commutative ring, and let {A1,...,An} be a pairwise comaximal set
ofn ideals. Prove that A1 ···An = A1 ∩ ··· ∩ An. (Hint: recall that
A1 ···An ⊆ A1 ∩···∩An from 8.3.8).
True or False
The magnetic field at a radial distance R from a current element i
dl is a maximum when the current element direction is at right
angles to the radial line.
Induced electric field lines are continuous, i.e. having no
beginning or ending points.
An arbitary shaped Amperian loop through which no currents pass
may itself pass through a magnetic field.
Induced electric field lines have a beginning and end point.
The negative sign in Faraday's Law (Lenz's...
What should a firm do if the marginal product obtained from the
last dollar spent on capital is smaller than the marginal product
derived from the last dollar spent on labor and why?