If F is a free module over a ring with identity such that F has a basis

Question:

If F is a free module over a ring with identity such that F has a basis of finite cardinality n ≥ 1 and another basis of cardinality n + 1, then F has a basis of cardinality m for every m ≥ n (m ϵ N*).

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 03:04 AM

See More Questions

If one attempted to dualize the notion of free module over a ring R (and called the object so defined "co-free") the definition would read: An R-module F is co-free on a set X if there exists a...
Let R be a ring with identity. Show that R is not a free module on any set in the category of all R-modules (as defined in Exercise 2). Data from in exercise 2 Let R be any ring (possibly without...
Let R be any ring (possibly without identity) and Fa free R-module on the set X, with L : X F, as in Exercise 2. Show that L (X) is a set of generators of the R-module F. is commutative. Conclude...
What are the institutional features that appear to have delayed the implementation of IFRS in India?
Admitting all exceptions to the rule, the rule is that free trade benefits all nations. Make the case.
On December 30, 2012, KPMG signs a $150,000 contract to provide accounting services to one of its clients in 2013 . KPMG has a December 31 year-end. Which accounting principle or assumption requires...
What would happen to the NPV of the above project if the inflation rate was expected to be 4 percent in each of the next four years? You may use either LO8 Excel or CB to determine your answer.
Define each of the following concepts, and then operationally define each one by providing correspondence rules between the definition and the scale: a. A good bowler b. The television audience for...
It's not the crown jewels, but it's certainly the family silver they're selling," was the verdict of one retail analyst on Pinault- Printemps-Redoute's decision to negotiate the sale of its Finaref...
The text used Poissons equation to show that the charge density of a point electric dipole with moment p located at the point r 0 is D (r) = p (r r 0 ). (a) Derive the given formula for D (r)...
Proof that Let K be a ring with identity and Fa free K-module with an infinite denumerable basis {e 1 ,e 2 , ... }. Then R = Hom K (F,F) is a ring by Exercise 1. 7(b). If n is any positive integer,...
Assume R has an identity. Let be the category of all unitary R-R bimodules and bimodule homomorphisms (that is, group homomorphisms : A B such that (ras) = r(a)s for all r,s R). Let X = { 1 R }...
In one cycle, a heat engine takes in 900 J of heat from a high-temperature reservoir and releases 600 J of heat to a lower-temperature reservoir. a. How much work is done by the engine in each cycle?...
Based on contract law principles, do you think the jury\'s verdict against the Loewen Group for $ 5 0 0 million was appropriate? Why or why not? What factors should the jury have considered in...
5.) Consider you have two systems - one filled with (1kg) water and the other with (1kg) of air. Both systems are at 1000 kPa and 30 C. Determine numerically which fluid system has the larger...
Question 3: The partnership of Blossom, Blue, and Kingbird engaged you to adjust its accounting records and convert them uniformly to the accrual basis in anticipation of admitting Kerns as a new...
Instructions : Build an Excel spreadsheet using the accounting equation (Assets = Liabilities + Shareholders' Equity). Remember that each transaction has an equal effect on both the left-hand side...
7.3 Fill in the spreadsheet below to calculate the port- folio return and risk between Zenon and Dynamics, given the 10 years of annual returns for each stock and portfolio weights of 50/50. (a) How...
How have trade balances in Asia contributed to the cause of the current Asian crisis?
The sales department of P. Gillen Manufacturing Company has forecast sales in March to be 20,000 units. Additional information follows: Finished goods inventory, March 1 . . . . . . . . . . . . . . ....
Researchers often use z tests to compare their samples to known population norms. The Graded Naming Test (GNT) asks respondents to name objects in a set of 30 black-and-white drawings. The test,...
In your own words, define the word effectfirst as you would use it in everyday conversation and then as a statistician would use it.
In an exercise in Chapter 7, we asked you to conduct a z test to ascertain whether the Graded Naming Test (GNT) scores for Canadian participants differed from the GNT norms based on adults in...
Jennifer purchased a home for $1,000,000 in 2016. She paid $200,000 cash and borrowed the remaining $800,000. This is Jennifer's only residence. Assume that in year 2024, when the home had...
business plan describing company with strengths and weaknesses. Any gaps in plan. Recommendations for improvement of the plan.
You wish to buy a car today for $35,000. You plan to put 10% down and finance the rest at 5.20% p.a. for six years. You will make equal monthly payments of $_______.

If F is a free module over a ring with identity such that F has a basis

Question:

Step by Step Answer:

Let R be a ring with identity and F be a free module over R with a basis of finite cardinality n 1 and another basis of cardinality n 1 We want to sho...View the full answer

Algebra Graduate Texts In Mathematics 73

Students also viewed these Mathematics questions