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:
Answer rating: 57% (7 reviews)
Let a b be nilpotent elements in a commutative ring R This means that there exist positive integers ...View the full answer
Answered By
Deepak Sharma
0.00
0 Reviews
10+ Question Solved
Related Book For
Algebra Graduate Texts In Mathematics 73
ISBN: 9780387905181
8th Edition
Authors: Thomas W. Hungerford
Question Posted:
Students also viewed these Mathematics 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.
-
a. Discuss the purpose of the cash budget. b. If the cash for the first quarter of the fiscal year indicates excess cash at the end of each of the first two months, how might the excess cash be used?
-
a. Define a material weakness. b. What are the auditor's responsibilities when a material weakness exists?
-
A methanol?water feed stream is introduced to a vaporizer in which a molar fraction f of the feed is vaporized. The feed has a methanol mole fraction of x F = 0.4, and the vaporizer operates at a...
-
Answer the following for the function f(x)=3x +5 Part A - Find the average rate of change from x = 1 to x = 4 Part B Find the instantaneous rate of change at x=3
-
XYZ is a calendar-year corporation that began business on January 1, 2017. For 2017, it reported the following information in its current year audited income statement. Notes with important tax...
-
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.
-
Keystone Company has two classes of capital stock outstanding: 10%, $40 par preferred and $1 par common. During the fiscal year ended November 30, 2013, the company was active in transactions...
-
When a Servlet is created, the init() method is called to do the initialization for the Web server. (True/False)
-
List five steps to build a data query from a Java database application project to a relational database using the Java runtime object method.
-
Explain why a complete ban on plastic bags would be inefficient. Plans to curtail use of plastic bags, but not much action Last summer, Seattle approved a 20-cents charge on plastic shopping bags,...
-
One can run and test any Oracle package with a stored procedure directly in the Oracle SQL Developer environment. (True/False)
-
Explain how Chinas price controls have changed consumer surplus, producer surplus, total surplus, and the deadweight loss in the markets for coal, gasoline, and diesel. Chinese power plants have run...
-
To a 0.444 g sample of H2O, 23.4 J of heat are added. What is its change in temperature? The heat capacity of H2O is in Table 7.1.
-
d. The characteristic equation of a control system is given by s+2s+8s+12s+20s+16+16=0. Determine the number of the roots of the equation which lie on the imaginary axis of s-plane
-
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...
-
Tough Fluid Dynamics Problem attached. Problem doesn't give diameter of pipe so we don't know velocity. The best approach would probably be to work in flow rates instead of velocities. Maybe solve...
-
A company is implementing Dynamics 3 6 5 Supply Chain Management. During requirements gathering, the company determines that an independent software vendor ( ISV ) is needed for small parcel...
-
Florence wrote goals that are consistent with her personality and attitudes toward money management. Amy s desired financial outcomes are not expressed in monetary terms and are not prioritized....
Study smarter with the SolutionInn App