(a)
To find: The additive identity in the ring, the unity element, and the additive and multiplicative inverse of
(a)
Answer to Problem 11E
The additive and multiplicative inverse of
Explanation of Solution
Given Information:
R is an equivalence relation on
It is given that Element
So,
Hence, the additive identity is
Now
Therefore, additive inverse is
For additive inverse of
So,
This leads to,
As,
For the multiplicative inverse first we have to determine the multiplicative identity
This will give us
Multiplicative inverse of
Hence, the multiplicative inverse of
(b)
To Prove:
(b)
Explanation of Solution
Given Information:
Given that
Prove:
It is given that
So,
Thus, it is a ring homomorphism
Hence, proved.
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Chapter 6 Solutions
A Transition to Advanced Mathematics
- Let A=R0, the set of all nonzero real numbers, and consider the following relations on AA. Decide in each case whether R is an equivalence relation, and justify your answers. (a,b)R(c,d) if and only if ad=bc. (a,b)R(c,d) if and only if ab=cd. (a,b)R(c,d) if and only if a2+b2=c2+d2. (a,b)R(c,d) if and only if ab=cd.arrow_forwarda. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the equivalence class [ 3 ]. b. Let R be the equivalence relation congruence modulo 4 that is defined on Z in Example 4. For this R, list five members of equivalence class [ 7 ].arrow_forward5. Let be the relation “congruence modulo ” defined on as follows: is congruent to modulo if and only if is a multiple of , we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .arrow_forward
- 4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .arrow_forward34. Let be the set of eight elements with identity element and noncommutative multiplication given by for all in (The circular order of multiplication is indicated by the diagram in Figure .) Given that is a group of order , write out the multiplication table for . This group is known as the quaternion group. (Sec. Sec. Sec. Sec. Sec. Sec. Sec. ) Sec. 22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. 32. Find the centralizer for each element in each of the following groups. a. The quaternion group in Exercise 34 of section 3.1 Sec. 2. Let be the quaternion group. List all cyclic subgroups of . Sec. 11. The following set of matrices , , , , , , forms a group with respect to matrix multiplication. Find an isomorphism from to the quaternion group. Sec. 8. Let be the quaternion group of units . Sec. 23. Find all subgroups of the quaternion group. Sec. 40. Find the commutator subgroup of each of the following groups. a. The quaternion group . Sec. 3. The quaternion group ; . 11. Find all homomorphic images of the quaternion group. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined byarrow_forward[Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and multiplication are defined as follows: if and only if and in , Given that is a ring, determine whether is commutative and whether has a unity. Justify your decisions. [Type here]arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,