An element of a ring is nilpotent if a n = 0 for some n. Prove that

Question:

An element of a ring is nilpotent if aⁿ = 0 for some n. Prove that in a commutative ring a + b is nilpotent if a and b are. Show that this result may be false if R is not commutative.

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

In a ring R the following conditions are equivalent. (a) R has no nonzero nilpotent elements (see Exercise 12). (b) If a R and a 2 = 0, then a = 0. Data from exercise 12 An element of a ring is...
Let R be a commutative ring and N an ideal of R. Referring to Exercise 30, show that if every element of N is nilpotent and the nilradical of R/N is R/N, then the nilradical of R is R. Data from...
Referring to Exercise 30, show that if N is the nilradical of a commutative ring R, then R/N has as nilradical the trivial ideal {0 + N}. Data from Exercise 30 An element a of a ring R is nilpotent...
Cubitene is a diterpene present in the defense secretion of a species of African termite. What unusual feature characterizes the joining of isoprene units in cubitene?
The creation of unwanted inventories or the depletion of wanted inventories signals coming changes in the level of national income. Explain.
Levitt Corporation manufactures a childs car seat produced in three separate departments: Molding, Assembling, and Finishing. It uses the weighted-average process-costing method to account for costs...
10-3. What are the five categories of product adopters in the diffusion of innovations?
Mike and Julie Bedard are a working couple. They will file a joint income tax return. This year, they have the following taxable income: 1. $125,000 from salary and wages (ordinary income) 2. $1,000...
1. PANDA Paper Industry Co. uses the discounted payback period method to evaluate investments in capital assets. The company expects the following annual cash flows from an investment of $4,500,000:...
A rotating step shaft is loaded as shown, where the forces F A and FB are constant at 600 lbf and 300 lbf, respectively, and the torque T alternates from 0 to 1800 lbf in. The shaft is to be...
Let R be a Euclidean domain with associated function : R - {0} N. a = qob +r, with r=0 b = qrr2, with r = 0 92r2r3, with r3 = 0 r = or (r) < y(b); or_y(r) < y(ri); or (r3) < (r); _rk =qk+rk+1+rk+2,...
A commutative ring with identity is local if and only if for all r, s R, r + s = 1 R implies r or s is a unit.
Negative Growth Antiques R Us is a mature manufacturing firm. The company just paid a 12 dividend, but management expects to reduce the payout by 4 per cent per year indefinitely. If you require an...
What is an incident in which a famous person wore or used a product (not as part of a paid endorsement or ad) and it caused a buying frenzy. Explain how the manufacturer or service provider reacted
What is a "heavyweight project team" and how does it differ from the traditional approach used for organizing development projects at Eli Lilly?This consists of two issues:First, an evaluation of the...
Consider the closed-loop system shown in Figure P11.6, where the transfer function of the process is that of a second-order system, i.e. k Ts +25TS +1 G,(s)= Y sp(s) E(s) U(s) Y(s) Ge(s) Gp(s) Figure...
1. Do you feel we have come along way with inventory in 10 years? 2. How did COVID affect the supply chain in your current hospital? Were any of the inventory systems/topics used, or relevant or...
Identify at least one way in which your writing skills have improved this semester and reflect on how you might use this skill in your career. You can include research, presentation, and report...
What role do you think empowerment plays at L.L. Bean? Because the companys retail Web site has proven to be so successful, L.L. Bean recently announced the closing of one of its four call centers,...
What are the risks and liability factors in an audit? What are the implications to the auditor? What are the implications to the organization? How can the auditor mitigate these risks and liability...
Determine the actual percent of t-values that exceed 2.776 in absolute value. Do you feel the central-limit theorem is applicable to these data for samples of size 5? S I M U L A T I O N Nutrition On...
Suppose two total cholesterol determinations are obtained from one child, yielding an average value of 200 mg/dL. What is a two-sided 90% CI for the true mean total cholesterol for that child?...
Suppose an average of two total cholesterol determinations is to be used as a screening tool to identify children with high cholesterol. The investigators wish to find a value c, such that all...
Portfolio return and beta Personal Finance Problem Jamie Peters invested $ 1 1 3 , 0 0 0 to set up the following portfolio one year ago: a . Calculate the portfolio beta on the basis of the original...
. Emerson Cammack wishes to purchase an annuity contract that will pay him $7,000 a year for the rest of his life. The Philo Life Insurance Company figures that his life expectancy is 20 years, based...
Integrity Inc. can sell 20-year, $1,000 par value bonds paying semi-annual interests with a 10% coupon. The bonds can be sold for $1,050 each; flotation cost of $50 per bond will be incurred in this...

An element of a ring is nilpotent if a n = 0 for some n. Prove that

Question:

Step by Step Answer:

Let a b be nilpotent elements in a commutative ring R This means that there exist positive integers ...View the full answer

Algebra Graduate Texts In Mathematics 73

Students also viewed these Mathematics questions