Let S be the ring of all n X n matrices over a division ring D. (a)

Question:

Let S be the ring of all n X n matrices over a division ring D.

(a) S has no proper ideals (that is, 0 is a maximal ideal). 8 or argue directly, using the matrices E_r,_s mentioned there.]

(b) S has zero divisors. Consequently,

(i) S ≅ S/0 is not a division ring and

(ii) 0 is a prime ideal which does not satisfy condition (1).

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Algebra Graduate Texts In Mathematics 73

ISBN: 9780387905181

8th Edition

Authors: Thomas W. Hungerford

See More Books

Question Posted: Apr 15, 2023 03:05 AM

See More Questions

Let : R S be an epimorphism of rings with kernel K. (a) If P is a prime ideal in R that contains K, then f(P) is a prime ideal in S [see Exercise 13]. (b) If Q is a prime ideal in S, then -1 ( Q)...
Let R be a ring with identity and S the ring of all n X n matrices over R. J is an ideal of S if and only if J is the ring of all n X n matrices over I for some ideal I in R.
(a) The center of the ring S of all 2 X 2 matrices over a field F consists of all matrices of the form (b) The center of Sis not an ideal in S. (c) What is the center of the ring of all n X n...
A railroad flatcar is traveling to the right at a speed of 13.0 m/s relative to an observer standing on the ground. Someone is riding a motor scooter on the flatcar (Fig. 3.43). What is the velocity...
Professor Kangoh Lee asks his students at Towson State University, "Suppose that in an economy, real consumption, real investment, and real government purchases remain the same from one year to...
As amanager, what would you focus on to improve the profitability of Spirit Beverages?
11-9. What are the three steps in setting a final price?
Employees at your company disagree about the accounting for sales returns. The sales manager believes that granting more generous return provisions and allowing customers to order items on a bill and...
Geeta has completed her B.Tech. In Chemical Engineering from a famous Indian Institute of Technology. She wants to start a chemical production unit wherein such chemicals will be produced which are...
River Spray Company was organized to grow cranberries. They entered into an agreement with a landowner to lease 125 acres to develop a cranberry bog. The agreement states that River Spray will be...
If R = Z, A 1 = (6) and A 2 = (4), then the map : R/ A 1 A 2 R/ A 1 X R/ A 2 of Corollary 2.27 is not surjective.
Let R be a commutative ring with identity, I an ideal of R and : R R/ I the canonical projection. (a) If S is a multiplicative subset of R, then S = (S) is a multiplicative subset of R/ I. (b) The...
Using L(f) in Prob. 10, find L(f 1 ), where f 1 (t) = 0 if t < 2 and f1(t) = 1 if t > 2. Data from Prob. 10 Find the transform. Show the details of your work. Assume that Î±, b, Ï,...
The University of Cincinnati Center for Business Analytics is an outreach center that collaborates with industry partners on applied research and continuing education in business analytics. One of...
For a data set of the pulse rates for a sample of adult females, the lowest pulse rate is 38 beats per minute, the mean of the listed pulse rates is x = 78.0 beats per minute, and their standard...
A student earned grades of A, C, B, A, and D. Those courses had these corresponding numbers of credit hours: 5, 3, 4, 3, and 2. The grading system assigns quality points to letter grades as follows:...
Ch 3: Forecasting: Tracking Signals, Mad, Exponential Smoothing, Control Charts Media Consultants (10 Pts). Media Consultants uses proven techniques to measure forecast accuracy and to determine when...
Question 2 What is the energy (in joules) of the photon absorbed by a hydrogen atom to cause a ground-state electron to move to the n = 3 energy level? Record your answer in scientific notation to 3...
On January 1, 2016, Brooke Daniels who is single started a new job. She was eligible for health insurance but, decided not to take the insurance offered by her employer. She did not require any...
What is a make-or-buy decision?
Suppose that four cases are observed over 1 year among 300 girls 04 years of age with smoking mothers. Answer Problem 7.17 based on these data? Pulmonary Disease Suppose the annual incidence of...
What is the SMR for this group? Occupational Health The mortality experience of 8146 male employees of a research, engineering, and metal-fabrication plant in Tonawanda, New York, was studied from...
Perform a significance test to assess whether there is an association between long duration of employment and mortality from cirrhosis of the liver in the group hired prior to 1946. Report a p-value?...
A zero-coupon bond bears a higher interest rate risk than a coupon-paying bond, given other bond characteristics are equal. True False
Company management decided to restructure its balance sheet. Current long term debt of 8 mio euros will be increased to 20 mio euros. Interest for the debt is 4%. Borrowed 12 mio euros will used to...
Susan loans 10000 to Jim. Jim repays the loan with yearly instalments at the end of each year. Interest is expected to be 5% the first five years, and 10% the last five years. Calculate the...

Let S be the ring of all n X n matrices over a division ring D. (a)

Question:

Step by Step Answer:

a Let I be a nonzero ideal of S Then there exists a nonzero matrix A aij I Without loss of ge...View the full answer

Algebra Graduate Texts In Mathematics 73

Students also viewed these Mathematics questions