Let R be a ring without identity and with no zero divisors Let S be the ring whose additive group is R X Z as in the proof of Theorem 1 10 Let A (r,n) S rx nx O for every x R (a) A is an ideal in S (b) S A has an identity and contains a subring isomorphic to R (c) S A has no zero divisors ...

a To show that A is an ideal of S we need to verify that for any rn and sm in A and any xy in S we h ...

Let R be a ring without identity and with no zero divisors. Let S be the ring

Question:

Let R be a ring without identity and with no zero divisors. Let S be the ring whose additive group is R X Z as in the proof of Theorem 1.10. Let A= {(r,n) ϵ S|rx+nx = O for every x ϵ R}.

(a) A is an ideal in S.

(b) S/ A has an identity and contains a subring isomorphic to R.

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 16, 2023 03:56 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)...
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...
Joe Greatan was a real estate salesman. Greatan sold to Darman, Doden & Droffats Associates (Triple D), a general partnership, an apartment building that needed extensive renovation. Greatan...
Consider the following velocity distribution curves A and B. a. If the plots represent the velocity distribution of 1.0 L of He(g) at STP versus 1.0 L of Cl2(g) at STP, which plot corresponds to each...
When firms increase dividends, stock prices tend to increase. One reason given for this price reaction is that dividends operate as a positive signal. What is the increase in dividends signaling to...
The following costs and inventory data were taken from the accounts of Winsto Company fo Inventories: Raw materials Work in process Finished goods Costs incurred: W 1-Jan-22 31-Dec-22 $8,000 $7,000...
Air flows in a 15-cm-diameter horizontal pipe. At section 1, $p=600 \mathrm{kPa}, T=70{ }^{\circ} \mathrm{C}$, and $V=35 \mathrm{~m} / \mathrm{s}$. At section 2: $T=42^{\circ} \mathrm{C}$ and...
No coffee is produced in the United States. Americans can buy as much coffee as they want from foreign producers at a price of $10 per pound. At this price, they buy 1,000 pounds per week. The U.S....
Thomas Company began its operations in 2 0 2 7 and applies overhead costs to jobs using a pre - determined overhead rate equal to 6 0 % of direct labor cost. During 2 0 2 7 , Thomas Company began...
Consider again the CHB Inc. problem described in Problem 10. Suppose only a limted number of PPBs can be placed. CHB would like to place this limited number of PPBs in counties so that the allowable...
Let R be a ring without identity. Embed Rina ring S with identity and characteristic zero as in the proof of Identify R with its image in S. (a) Show that every element of S may be uniquely expressed...
Let R be a ring with more than one element such that for each nonzero a R there is a unique b R such that aba = a. Prove: (a) R has no zero divisors. (b) bab = b. (c) R has an identity. (d) R is a...
Hicks Contracting collects and analyzes cost data in order to track the cost of installing decks on new home construction jobs. The following are some of the costs that they incur. Classify these...
Massive Dynamic Computers sources the components for its laptops from various suppliers on the market. The firm pays $100 for processors, $35 for disk drives, $50 for screens, $10 for memory, and $40...
How much potential energy due to gravity a person have if they were standing on top of a building that is 5 2 . 9 M high assume that they have a mass of 9 4 . 5 Kg?
Given the Following Information for 2 0 2 3 Z owns 7 0 % of Y Y net income for 2 0 2 3 is $ 1 5 0 , 0 0 0 Y declares dividends of $ 1 0 , 0 0 0 during 2 0 2 3 . Y has 2 0 , 0 0 0 shares of $ 1 par...
What is Pros and Cons of a business memo? Summarize one advantage and one disadvantage of memos. Use reference: Smith, J. (2015). 7.2: Memos - Communication at Work . Pressbooks....
1. A mechanic lifts a 1740 kg car using a hydraulic lift. The lift uses two pistons with diameters in ratio of d/d = 5.30, with the car lifted using the large piston to a height of 1.75 m. (a) How...
A researcher studying violent behavior in elementary school children asks the children's parents how much time each child spends playing computer games and has their teachers rate each child on the...
r = 0.18 Find the coefficients of determination and non-determination and explain the meaning of each.
Use ANOVA methods to compare the mean change in serum-creatinine values from the baseline visit to the 1978 visit among women in the high-NAPAP group, low-NAPAP group, and the control group?
Compare your results in Problems 12.44 and 12.45 with the corresponding results in Table 12.9 (on page 582) for MAXFWT and Problem 12.40 for IQF where parametric methods were used. Table 12.9:...
Answer Problem 12.44 for IQF (full-scale IQ)?
What is the difference between the leverage ratio (LR) and the loan-to-value ratio (LTV)? How much greater property value can be purchased with a 75% LTV than with a 50% LTV? What is the LR...
Onshore Bank has $50 million in assets, with risk-adjusted assets of $30 million. Tier I capital is $2 million, and Tier II capital is $1.5 million. How will the following transactions affect the...
A competitor had offered $35,000 cash for machine, an amount that represented its current book value. If Davidson opted to the keep the machine, Magic would continue to claim depreciation of $6000...

Let R be a ring without identity and with no zero divisors. Let S be the ring

Question:

Step by Step Answer:

a To show that A is an ideal of S we need to verify that for any rn and sm in A and any xy in S we h...View the full answer

Algebra Graduate Texts In Mathematics 73

Students also viewed these Mathematics questions