Question: Prove that a ring R is commutative if and only if (a + b)2 = a2 + 2ab + b2, for all a, b

Prove that a ring R is commutative if and only if (a + b)2 = a2 + 2ab + b2, for all a, b ∈ R.

Step by Step Solution

★★★★★

3.37 Rating (156 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

R commutative ba ab for ... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Document Format (1 attachment)

Word file Icon

954-M-L-A-L-S (8454).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Q:

Let (S, +, ) and (T, +', ') be two rings. For R = S T, define addition "" and multiplication "" by (s1, t1) (s2, t2) = (S1 + S2, t1 +' t2), (s1 t1) (s2, t2) = (S1 s2, t1. t2). (a) Prove that...

Q:

A ring R is called Boolean if a2 = a for all a R. If R is Boolean, prove that (a) a + a = 2a = z, for all a R ', and (b) R is commutative.

Q:

Prove that multiplication of complex numbers is commutative, as stated at the beginning of Sec. 2.

Q:

Compute the following double sum: (a) X 3 i=1 X 2 j=1 (i j) (b) X 3 i=1 X 2 j=0 j (c) X 2 i=0 X 3 j=0 i 2 j 3 A= 0 1 7 2 4 3 , B= 4 1 2 3 0 2 0 3 4 1 True or False The following is true for all 2 2...

Q:

Vispirms aplkosim piemru ( sin + cos )2 - sin2 = 1 pacelsim iekavu kvadrt, izmantojot formulu (a + b )2 = a2 + 2ab + b2 sin2 + 2sincos + cos2 - 2sincos = 1, sin2 tika prveidots saska ar divkr...

Q:

Evaluate the polynomial for a = 4 and b = 2. a 2 + 2 ab b 2

Q:

lJ lJLJlJ Mr. and Mrs. Garcia have a total of $100,000 to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of 18%lyear, while the bonds and the money market...

Q:

H. W Exercise 8. Prove that a ring (R,+..) is commutative if and only if (a+b)2 = a + 2(a.b) + b, for all a, b ER. 11. Austin Blake borrowed $12,000 on a 11% 240-day note. On the 90th day, he paid...

Q:

12:08 I Georgia Standards Of Excellence Sec 1.5 -Revisiting Quadratics Solving Quadratics By Factoring Name: 1. (REVIEW) FACTOR the following if possible (not all may be factored): a. x -10x -24 b. w...

Q:

9/14/10 12:39 PM Page 381 Chapter dug84356_ch06a.qxd 6 Rational Expressions Advanced technical developments have made sports equipment faster, lighter, and more responsive to the human body. Behind...

Q:

Analyze how to create value and sustained competitive advantage through business strategy. Business strategy Competitive and cooperative strategies that emphasize improvement of the competitive...

Q:

Find bases for each of the four fundamental subspaces of 3297 1181 2246 1215 1123

Q:

A survey is done to estimate the proportion of U.S. adults who think that cell phone use while driving should be illegal. In the survey, 54% of a randomly selected sample of 1025 individuals said...

Q:

In 1998, Johnson Controls, Inc. (JCI), began buying auto parts from Q.C. Onics Ventures, LP. For each part, JCI would inform Onics of its need and ask the price. Onics would analyze the...

Q:

Find the height, width, area, and distance specified. a. Total height of the tree b. Width of the lake c. Area of the triangle d. Distance from the boat to the lighthouse Lake go 68.4 m 29 48 em AT...

Q:

Find a polynomial equation of least degree with integer coefficients that has roots -3 and

Q:

Give parametric equations and a single equation using only x and y for the ellipse at right. 3.910 2-

Q:

The table above shows the rate of return (APR) for four investment alternatives. Which offers the highest EAR? A. Investment A B. Investment B C. Investment C D. Investment D

Q:

Hello! I need help with these. Complete and clear solution is needed. Thank you Activity #1: MEAN, MEDIAN, MODE - UNGROUPED Instruction A: Use the given data sets to answer the items in this section....

Q:

Apple Inc., a U.S. based corporation, wants to purchase Technology Inc. a company based in Singapore. The current U.S. Treasury bond rate (Rus) is 4%. The expected inflation rate in the United States...