An element a of a ring R is idempotent if a 2 = a. a. Show that

Question:

An element a of a ring R is idempotent if a² = a.

a. Show that the set of all idempotent elements of a commutative ring is closed under multiplication.

b. Find all idempotents in the ring Z₆ x Z₁₂.

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

Step by Step Answer:

Related Book For book-img-for-question

A First Course In Abstract Algebra

ISBN: 9780201763904

7th Edition

Authors: John Fraleigh

See More Books

Question Posted: Jan 04, 2023 05:05 AM

See More Questions

An element a of a ring R is idempotent if a 2 = a. Show that a division ring contains exactly two idempotent elements.
An element a of a ring R is nilpotent if a n = 0 for some n Z + . Show that the collection of all nilpotent elements in a commutative ring R is an ideal, the nilradical of R.
An element a of a ring R is nilpotent if a n = 0 for some n Z + . Show that if a and b are nilpotent elements of a commutative ring, then a + b is also nilpotent.
A process is controlled with a fraction nonconforming control chart with three-sigma limits, n = 100, UCL = 0.161, center line = 0.080, and LCL = 0. (a) Find the equivalent control chart for the...
More freshmen: For the data in Exercise 6: a. Construct a frequency polygon. b. Construct a relative frequency polygon. c. Construct a frequency o give. d. Construct a relative frequency o give.
Write a database description for each of the relations shown, using SQL DDL (shorten, abbreviate, or change any data names, as needed for your SQL version). Assume the following attribute data types:...
Pattys Pumpkin Pies collected the following production information relating to Novembers baking operations: Requirements: Complete the first two steps in the process costing procedure: 1. Summarize...
How does Fords Sync contribute to its competitive advantage? Is this a sustainable advantage? The old phrase, The bigger they are, the harder they fall, perfectly describes what has happened to the...
6. Given below are the oxides : Na2O, As2O3, N2O, NO and ChO7 Number of amphoteric oxides is: (a) 0 (b) 1 7. Match List -I with List - II: List-I List-II (A) Sphalerite (I) FeCO3 (B) Calamine (II)...
Jimmy owns a garden in which he has planted N trees in a row. After a few years, the trees have grown up and now they have different heights. Jimmy pays much attention to the aesthetics of his...
Partition the following collection of groups into subcollections of isomorphic groups. Here a * superscript means all nonzero elements of the set. Z under addition S 2 Z 6 R * under multiplication Z...
Show that the multiplication defined on the set F of functions in Example 18.4 satisfies axioms R2 and R3 for a ring. Data from 18.4 Example Let F be the set of all functions f: R R We know that (F,...
What is the main objective of the Dodd Frank Act?
External auditors play an important role in ensuring compliance with corporate governance principles through the protection of shareholders' interests. Critically evaluate the role of the external...
An environment with a small number of external elements where the elements are similar and the elements change frequently and unpredictably is a __________. complex-unstable environment...
Tina Muzinga is a single individual who is working on filing her tax return for the previous year. She has assembled the following relevant information: She received $27,000 from the sale of Disney...
1) Three commonly used productivity variables are: A) quality, external elements, and precise units of measure. B) labor, capital, and management. C) technology, raw materials, and labor. D)...
Hair Supply, Inc. paid $1,600,000 to acquire all of the common stock of Family Hair Corp. on January 1, 2XX0. Family Hair's reported earnings for 2XX0 totaled $396,000, and it paid $132,000 in...
If o is an outer measure and Ao is the -field of o- measurable sets, then show that Ao is complete (with respect to o).
You purchase a bond with a coupon rate of 6.7 percent, a par value $1,000, and a clean price of $905. Assume a par value of $1,000. If the next semiannual coupon payment is due in two months, what is...
A student scored 90 on the midterm exam, 78 on the final exam, and 83 on the term project. If these three components are weighted at 35%, 45%, and 20%, respectively, what is the weighted mean for her...
Provide a real or hypothetical example of a situation where the range could be misleading as a measure of dispersion.
A Polk Company survey found that households with someone who frequently plays golf have a median income of $54,826, compared to a median of $37,354 for all households. Under the assumption that the...
Problem 2: Pat and Chris Broderick have just purchased a home in Lincoln Park. The home's sale price was $549,000. As a veteran, Pat was able to get a 30-yr loan, with an APR of 3.75% compounded...
Summarize: The challenge of sustainability will impact the hospital health care setting significantly. Hospitals are high energy consumers due to their 24/7 operation. To become sustainable, they...
What are the four critical areas in which employees should have pride, enjoyment, trust, and benefit to enable high morale?

An element a of a ring R is idempotent if a 2 = a. a. Show that

Question:

Step by Step Answer:

a If a 2 a and b 2 b and if the ring is commutative then ab 2 abab aabb a ...View the full answer

A First Course In Abstract Algebra

Students also viewed these Mathematics questions