i need the answer quickly ۱:۳۱|TrueFalse* (10Let (R, +) be commutative ring with identity. Then a · b = a·c + b = c if and only if Rhas no zero divisors.TrueFalseملاحظة: اضغط على التالی ليظهر لك السؤالالثالثمحو النموذجالتاليرجوععدم إرسال كلمات المرور عبر نماذج Go ogle مطلقا.تم إنشاء هذا النموذج داخل University. Of Misanالإبلاغ عنإساءة الاستخدامGoogle zilai

Answered: * (10 Let (R, +,) be commutative ring…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 16E: a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the...

See similar textbooks

Similar questions

[Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]
11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.
44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.
[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
40. Let be idempotent in a ring with unity. Prove is also idempotent.
46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

* (10 Let (R, +,) be commutative ring with identity. Then a - b = a-c-b=c ifand only if R has no zero divisors. True O False