Abstract Algebra Answers

Compute the lower nilradical of the ring Tn(k) of upper triangular matrices over any ring k.

For a triangular ring T =
R M
0 S
(where M is an (R, S)-bimodule), show that Nil∗(T) =
Nil∗(R) M
0 Nil∗(S)

Let I be a nil left ideal of a ring R. Show that the set of matrices in Mn(R) whose kth column consists of elements of I and whose other columns are zero is a nil left ideal of Mn(R).

For any ring R, show that Nil*Mn(R) = Mn(Nil*R).

Let R[T] be a polynomial ring over R, and let N = R ∩ rad R[T]. Show that N is a semiprime ideal of R

When writing an equation in function form like. -5y-30x=20 I get -5y=-30x+20 how do I get the -5 factored out?

a hemispherical dome of a building contains 56 4/7 m3 of air . Find the radius of the dome

a) For the quadratic x^2-cx+d, with c, d integers and roots alpha & alpha-hat which are not integers. Establish that if |c|+|d| is less than or equal to 2, then Z[alpha] is a Euclidean Domain.

b) Find all such Z[alpha]'s satisfying the conditions of (a)

Last year, General hospital discharged a total of 19,231 patients. Its occupancy rate was 83%. Total operating expenses were $101 million. The average cost per discharge in the industry was $5,446. Based on this information, General hospital's average margin per discharge was:
a. worse than the average
b. better than the average
c. equal to the average
d. cannot compute from information provided