Problem 1TFE: True or False
Label each of the following statements as either true or false.
1. Every ring is an... Problem 2TFE: True or False Label each of the following statements as either true or false. Let R be a ring. The... Problem 3TFE: True or False Label each of the following statements as either true or false. Let R be a ring. Then... Problem 4TFE: True or False
Label each of the following statements as either true or false.
4. Both , the set of... Problem 5TFE: True or False Label each of the following statements as either true or false. If one element in a... Problem 6TFE: True or False Label each of the following statements as either true or false. Let x and y be... Problem 7TFE: True or False Label each of the following statements as either true or false. Let R be a ring with... Problem 8TFE: True or False
Label each of the following statements as either true or false.
8. A unity exists in... Problem 9TFE: True or False Label each of the following statements as either true or false. Any ring with unity... Problem 10TFE: True or False Label each of the following statements as either true or false. n is a subring of ,... Problem 1E: Exercises
Confirm the statements made in Example 3 by proving that the following sets are subrings... Problem 2E: Exercises
2. Decide whether each of the following sets is a ring with respect to the usual... Problem 3E: Exercises
3. Let Using addition and multiplication as they are defined in Example 5, construct... Problem 4E Problem 5E: Exercises
5. Let Define addition and multiplication in by and . Decide whether is a ring with... Problem 6E: Exercises Work exercise 5 using U=a. Exercise5 Let U=a,b. Define addition and multiplication in P(U)... Problem 7E: Exercises Find all zero divisors in n for the following values of n a. n=6 b. n=8 n=10 d. n=12 n=14... Problem 8E: Exercises
8. For the given values of , find all the units in
b.
d.
f. a prime... Problem 9E: Exercises Prove Theorem 5.3:A subset S of the ring R is a subring of R if and only if these... Problem 10E: Exercises
10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these... Problem 11E: Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the... Problem 12E: 12. (See Example 4.) Prove the right distributive law in:
.
Example 4 For, let denote the... Problem 13E: 13. Complete the proof of Theorem by showing that for any in a ring .
Theorem Zero Product
... Problem 14E: Let R be a ring, and let x,y, and z be arbitrary elements of R. Complete the proof of Theorem 5.11... Problem 15E: 15. Let and be elements of a ring. Prove that the equation has a unique solution.
Problem 16E: 16. Suppose that is an abelian group with respect to addition, with identity element Define a... Problem 17E: If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R. Problem 18E: 18. Find subrings and of such that is not a subring of .
Problem 19E: 19. Find a specific example of two elements and in a ring such that and .
Problem 20E Problem 21E: 21. Define a new operation of addition in by with a new multiplication in by.
a. Verify that... Problem 22E: 22. Define a new operation of addition in by and a new multiplication in by.
a. Is a... Problem 23E: Let R be a ring with unity and S be the set of all units in R. a. Prove or disprove that S is a... Problem 24E: Prove that if a is a unit in a ring R with unity, then a is not a zero divisor. Problem 25E Problem 26E Problem 27E: Suppose that a,b, and c are elements of a ring R such that ab=ac. Prove that is a has a... Problem 28E Problem 29E: 29. For a fixed element of a ring , prove that the set is a subring of.
Problem 30E Problem 31E: Let R be a ring. Prove that the set S={ xRxa=axforallaR } is a subring of R. This subring is called... Problem 32E: 32. Consider the set .
a. Construct addition and multiplication tables for, using the... Problem 33E: Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition... Problem 34E: The addition table and part of the multiplication table for the ring R={ a,b,c } are given in Figure... Problem 35E: 35. The addition table and part of the multiplication table for the ring are given
in Figure... Problem 36E Problem 37E: 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero... Problem 38E: An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2(). Problem 39E: 39. (See Exercise 38.) Show that the set of all idempotent elements of a commutative ring is
... Problem 40E: 40. Let be idempotent in a ring with unity. Prove is also idempotent.
Problem 41E: 41. Decide whether each of the following sets is a subring of the ring. If a set is not a
... Problem 42E: 42. Let .
a. Show that is a commutative subring of.
b. Find the unity, if one... Problem 43E: 43. Let .
a. Show that is a noncommutative subring of .
b. Find the unity element,... Problem 44E: 44. Consider the set of all matrices of the form, where and are real
numbers, with the... Problem 45E Problem 46E: 46. Let be a set of elements containing the unity, that satisfy all of the conditions in
... Problem 47E Problem 48E Problem 49E: An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set... Problem 50E: 50. Let and be nilpotent elements that satisfy the following conditions in a commutative
... Problem 51E: Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and... Problem 52E: 52. (See Exercise 51.)
a. Write out the elements of and construct addition and... Problem 53E Problem 54E Problem 55E Problem 56E: Suppose R is a ring in which all elements x are idempotent-that is, all x satisfy x2=x. (Such a ring... format_list_bulleted
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