Show, by example, that for fixed nonzero elements a and b

Problem 52E If a, b, and c are elements of a ring, does the equation ax + b = c always have a solution x? If it does, must the solution be unique? Answer the same questions given that a is a unit.

Problem 1CE Exercise 1. This software finds all solutions to the equation x 2 + y 2 = 0 in Z p. Run your program for all odd primes up to 37. Make a conjecture about the the number of solutions in Z p (where p is a prime) and the form of p . Exercise 4 . This software determines the order of the group of units in the ring of 2 by 2 matrices over Z n (that is, the group GL (2, Z n )) and the subgroup SL(2, Z n ). Run the program for n = 2, 3, 5, 7, 11, and 13. What relationship do you see between the order of GL (2, Z n ) and the order of SL (2, Z n ) in these cases? Run the program for n = 16, 27, 25, and 49 . Make a conjecture about the relationship between the order of GL (2, Z n ) and the order of SL (2, Z n ) when n is a power of a prime. Run the program for n = 32 . (Notice that when you run the program for n = 32 the table shows the orders for all divisors of 32 greater than 1.) How do the orders the two groups change each time you increase the power of 2 by 1? Run the program for n = 27 . How do the orders the two groups change each time you increase the power of 3 by 1? Run the program for n = 25 . How do the orders the two groups change when you increase the power of 5 by 1? Make a conjecture about the relationship between |SL(2,Z p i )| and |SL(2,Z p i+1 )| . Make a conjecture about the relationship between |GL(2,Z p i )| and |GL(2,Z p i+1 )| . Run the program for n = 12, 15, 20, 21, and 30. Make a conjecture about the order of GL (2, Z n ) in terms of the orders of GL (2, Z s ) and GL (2, Z t ) where n = st and s and t are relatively prime. (Notice that when you run the program for st the table shows the values for st, s and t .) For each value of n is the order of SL (2, Z n ) divisible by n ? Is it divisible by n + 1? Is it divisible by n - 1? Exercise 5. In the ring Z n this software finds the number of solutions to the equation x 2 = -1. Run the program for all primes between 3 and 29. How does the answer depend on the prime? Make a conjecture about the number of solutions when n is a prime greater than 2. Run the program for the squares of all primes between 3 and 29. Make a conjecture about the number of solutions when n is the square of a prime greater than 2. Run the program for the cubes of primes between 3 and 29. Make a conjecture about the number of solutions when n is any power of an odd prime. Run the program for n = 2, 4, 8, 16, and 32. Make a conjecture about the number of solutions when n is a power of 2. Run the program for n = 12, 20, 24, 28, and 36. Make a conjecture about the number of solutions when n is a multiple of 4. Run the program for various cases where n = pq and n = 2 pq where p and q are odd primes. Make a conjecture about the number of solutions when n = pq or n = 2 pq where p and q are odd primes. What relationship do you see between the number of solutions for n = p and n = q and n = pq ? Run the program for various cases where n = pqr and n = 2 pqr where p , q and r are odd primes. Make a conjecture about the number of solutions when n = pqr or n = 2 pqr where p , q and r are odd primes. What relationship do you see between the number of solutions when n = p , n = q and n = r and the case that n = pqr ? Exercise 6. This software determines the number of solutions to the equation X2 = -I where X is a 2 x 2 matrix with entries from Z n and I is the identity. Run the program for n = 32 . Make a conjecture about the number of solutions when n = 2 k where k > 1 . Run the program for n = 3, 11, 19, 23, and 31 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 3 . Run the program for n = 27 and 49 . Make a conjecture about the number of solutions when n has the form p i where p is a prime of the form 4q + 3 . Run the program for n = 5, 13, 17, 29, and 37 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 1 . Run the program for n = 6, 10, 14, 22; 15, 21, 33, 39; 30, 42. What seems to be the relationship between the number of solutions for a given n and the number of solutions for the prime power factors of n ?

Problem 1E Give an example of a finite noncommutative ring. Give an example of an infinite noncommutative ring that does not have a unity.

Problem 2CE Let Zn[i] = { a+bi | a, b belong to Zn, i2=-1 } (the Gaussian integers modulo n ). This software finds the group of units of this ring and the order of each element of the group. Run the program for n = 3, 7, 11, and 23. Is the group of units cyclic for these cases? Try to guess a formula for the order of the group of units of Zn[i] as a function of n when n is a prime and n mod 4 = 3. Run the program for n = 9 and 27. Are the groups cyclic? Try to guess a formula for the order when n = 3k. Run the program for n = 5, 13, 17, and 29. Is the group cyclic for these cases? What is the largest order of any element in the group? Try to guess a formula for the order of the group of units of Zn[i] as a function of n when n is a prime and n mod 4 = 1. Try to guess a formula for the largest order of any element in the group of units of Zn[i] as a function of n when n is a prime and n mod 4 = 1. On the basis of the orders of the elements of the group of units, try to guess the isomorphism class of the group. Run the program for n = 25. Is this group cyclic? Based on the number of elements in this group and the orders of the elements, try to guess the isomorphism class of the group.

The ring {0, 2, 4, 6, 8} under addition and multiplication modulo 10 has a unity. Find it.

Problem 3CE This software determines the isomorphism class of the group of units of Zn[i]. Run the program for n = 5, 13, 17, 29, and 37. Make a conjecture. Run the program for n = 3, 7, 11, 19, 23, and 31. Make a conjecture. Run the program for n = 5, 25, and 125. Make a conjecture. Run the program for n = 13. Make a conjecture. Run the program for n = 3, 9, and 27. Make a conjecture. Run the program for n = 7 and 49. Make a conjecture. Run the program for n = 11 and 121. Make a conjecture. Make a conjecture about the case where n = pk where p is a prime and p mod 4 = 1. Make a conjecture about the case where n = pk where p is a prime and p mod 4 = 3.

Problem 3E Give an example of a subset of a ring that is a subgroup under addition but not a subring.

Problem 4CE Exercise 1. This software finds all solutions to the equation x 2 + y 2 = 0 in Z p. Run your program for all odd primes up to 37. Make a conjecture about the the number of solutions in Z p (where p is a prime) and the form of p . Exercise 4 . This software determines the order of the group of units in the ring of 2 by 2 matrices over Z n (that is, the group GL (2, Z n )) and the subgroup SL(2, Z n ). Run the program for n = 2, 3, 5, 7, 11, and 13. What relationship do you see between the order of GL (2, Z n ) and the order of SL (2, Z n ) in these cases? Run the program for n = 16, 27, 25, and 49 . Make a conjecture about the relationship between the order of GL (2, Z n ) and the order of SL (2, Z n ) when n is a power of a prime. Run the program for n = 32 . (Notice that when you run the program for n = 32 the table shows the orders for all divisors of 32 greater than 1.) How do the orders the two groups change each time you increase the power of 2 by 1? Run the program for n = 27 . How do the orders the two groups change each time you increase the power of 3 by 1? Run the program for n = 25 . How do the orders the two groups change when you increase the power of 5 by 1? Make a conjecture about the relationship between |SL(2,Z p i )| and |SL(2,Z p i+1 )| . Make a conjecture about the relationship between |GL(2,Z p i )| and |GL(2,Z p i+1 )| . Run the program for n = 12, 15, 20, 21, and 30. Make a conjecture about the order of GL (2, Z n ) in terms of the orders of GL (2, Z s ) and GL (2, Z t ) where n = st and s and t are relatively prime. (Notice that when you run the program for st the table shows the values for st, s and t .) For each value of n is the order of SL (2, Z n ) divisible by n ? Is it divisible by n + 1? Is it divisible by n - 1? Exercise 5. In the ring Z n this software finds the number of solutions to the equation x 2 = -1. Run the program for all primes between 3 and 29. How does the answer depend on the prime? Make a conjecture about the number of solutions when n is a prime greater than 2. Run the program for the squares of all primes between 3 and 29. Make a conjecture about the number of solutions when n is the square of a prime greater than 2. Run the program for the cubes of primes between 3 and 29. Make a conjecture about the number of solutions when n is any power of an odd prime. Run the program for n = 2, 4, 8, 16, and 32. Make a conjecture about the number of solutions when n is a power of 2. Run the program for n = 12, 20, 24, 28, and 36. Make a conjecture about the number of solutions when n is a multiple of 4. Run the program for various cases where n = pq and n = 2 pq where p and q are odd primes. Make a conjecture about the number of solutions when n = pq or n = 2 pq where p and q are odd primes. What relationship do you see between the number of solutions for n = p and n = q and n = pq ? Run the program for various cases where n = pqr and n = 2 pqr where p , q and r are odd primes. Make a conjecture about the number of solutions when n = pqr or n = 2 pqr where p , q and r are odd primes. What relationship do you see between the number of solutions when n = p , n = q and n = r and the case that n = pqr ? Exercise 6. This software determines the number of solutions to the equation X2 = -I where X is a 2 x 2 matrix with entries from Z n and I is the identity. Run the program for n = 32 . Make a conjecture about the number of solutions when n = 2 k where k > 1 . Run the program for n = 3, 11, 19, 23, and 31 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 3 . Run the program for n = 27 and 49 . Make a conjecture about the number of solutions when n has the form p i where p is a prime of the form 4q + 3 . Run the program for n = 5, 13, 17, 29, and 37 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 1 . Run the program for n = 6, 10, 14, 22; 15, 21, 33, 39; 30, 42. What seems to be the relationship between the number of solutions for a given n and the number of solutions for the prime power factors of n ?

Problem 4E Show, by example, that for fixed nonzero elements a and b in a ring, the equation ax =b can have more than one solution. How does this compare with groups?

Problem 5CE Exercise 1. This software finds all solutions to the equation x 2 + y 2 = 0 in Z p. Run your program for all odd primes up to 37. Make a conjecture about the the number of solutions in Z p (where p is a prime) and the form of p . Exercise 4 . This software determines the order of the group of units in the ring of 2 by 2 matrices over Z n (that is, the group GL (2, Z n )) and the subgroup SL(2, Z n ). Run the program for n = 2, 3, 5, 7, 11, and 13. What relationship do you see between the order of GL (2, Z n ) and the order of SL (2, Z n ) in these cases? Run the program for n = 16, 27, 25, and 49 . Make a conjecture about the relationship between the order of GL (2, Z n ) and the order of SL (2, Z n ) when n is a power of a prime. Run the program for n = 32 . (Notice that when you run the program for n = 32 the table shows the orders for all divisors of 32 greater than 1.) How do the orders the two groups change each time you increase the power of 2 by 1? Run the program for n = 27 . How do the orders the two groups change each time you increase the power of 3 by 1? Run the program for n = 25 . How do the orders the two groups change when you increase the power of 5 by 1? Make a conjecture about the relationship between |SL(2,Z p i )| and |SL(2,Z p i+1 )| . Make a conjecture about the relationship between |GL(2,Z p i )| and |GL(2,Z p i+1 )| . Run the program for n = 12, 15, 20, 21, and 30. Make a conjecture about the order of GL (2, Z n ) in terms of the orders of GL (2, Z s ) and GL (2, Z t ) where n = st and s and t are relatively prime. (Notice that when you run the program for st the table shows the values for st, s and t .) For each value of n is the order of SL (2, Z n ) divisible by n ? Is it divisible by n + 1? Is it divisible by n - 1? Exercise 5. In the ring Z n this software finds the number of solutions to the equation x 2 = -1. Run the program for all primes between 3 and 29. How does the answer depend on the prime? Make a conjecture about the number of solutions when n is a prime greater than 2. Run the program for the squares of all primes between 3 and 29. Make a conjecture about the number of solutions when n is the square of a prime greater than 2. Run the program for the cubes of primes between 3 and 29. Make a conjecture about the number of solutions when n is any power of an odd prime. Run the program for n = 2, 4, 8, 16, and 32. Make a conjecture about the number of solutions when n is a power of 2. Run the program for n = 12, 20, 24, 28, and 36. Make a conjecture about the number of solutions when n is a multiple of 4. Run the program for various cases where n = pq and n = 2 pq where p and q are odd primes. Make a conjecture about the number of solutions when n = pq or n = 2 pq where p and q are odd primes. What relationship do you see between the number of solutions for n = p and n = q and n = pq ? Run the program for various cases where n = pqr and n = 2 pqr where p , q and r are odd primes. Make a conjecture about the number of solutions when n = pqr or n = 2 pqr where p , q and r are odd primes. What relationship do you see between the number of solutions when n = p , n = q and n = r and the case that n = pqr ? Exercise 6. This software determines the number of solutions to the equation X2 = -I where X is a 2 x 2 matrix with entries from Z n and I is the identity. Run the program for n = 32 . Make a conjecture about the number of solutions when n = 2 k where k > 1 . Run the program for n = 3, 11, 19, 23, and 31 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 3 . Run the program for n = 27 and 49 . Make a conjecture about the number of solutions when n has the form p i where p is a prime of the form 4q + 3 . Run the program for n = 5, 13, 17, 29, and 37 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 1 . Run the program for n = 6, 10, 14, 22; 15, 21, 33, 39; 30, 42. What seems to be the relationship between the number of solutions for a given n and the number of solutions for the prime power factors of n ?

Problem 5E Prove Theorem 12.2. REFERENCE: Theorem 12.2 Uniqueness of the Unity and Inverses If a ring has a unity, it is unique. If a ring element has a multiplicative inverse, it is unique.

This software determines the number of solutions to the equation \(X^2 = -I\) where X is a 2 x 2 matrix with entries from \(Z_n\) and I is the identity. Run the program for n = 32 . Make a conjecture about the number of solutions when n = 2 k where k > 1 . Run the program for n = 3, 11, 19, 23, and 31 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 3 . Run the program for n = 27 and 49 . Make a conjecture about the number of solutions when n has the form \(p^i\) where p is a prime of the form 4q + 3 . Run the program for n = 5, 13, 17, 29, and 37 . Make a conjecture about the number of solutions when n is a prime of the form 4q + 1 . Run the program for n = 6, 10, 14, 22; 15, 21, 33, 39; 30, 42. What seems to be the relationship between the number of solutions for a given n and the number of solutions for the prime power factors of n? Go to https://www.d.umn.edu/~jgallian/compsciProject2018/html/chap12/ch12ex6.html

Problem 6E Find an integer n that shows that the rings Zn need not have the following properties that the ring of integers has. a. a 2 = a implies a = 0 or a = 1. b. ab = 0 implies a = 0 or b = 0. c. ab = ac and a?0 imply b = c. Is the n you found prime?

Problem 7E Show that the three properties listed in Exercise 6 are valid for Zp, where p is prime. REFERENCE: Find an integer n that shows that the rings Zn need not have the following properties that the ring of integers has. a. a 2 = a implies a = 0 or a = 1. b. ab = 0 implies a = 0 or b = 0. c. ab = ac and a?0 imply b = c. Is the n you found prime?

Problem 8E Show that a ring is commutative if it has the property that ab = ca implies b = c when a ? 0.

Problem 9E Prove that the intersection of any collection of subrings of a ring R is a subring of R.

Problem 10E Verify that Examples 8 through 13 in this chapter are as stated. REFERENCE:

Prove rules 3 through 6 of Theorem 12.1.

Problem 13E Describe all the subrings of the ring of integers.

Problem 15E Show that if m and n are integers and a and b are elements from a ring, then (m · a)(n · b) 5 (mn) · (ab). (This exercise is referred to in Chapters 13 and 15.)

Problem 12E Let a, b, and c be elements of a commutative ring, and suppose that a is a unit. Prove that b divides c if and only if ab divides c.

Problem 16E Show that if n is an integer and a is an element from a ring, then n · (–a) = –(n · a).

Problem 14E Let a and b belong to a ring R and let m be an integer. Prove that m · (ab) = (m · a)b = a(m · b).

Problem 17E Show that a ring that is cyclic under addition is commutative.

Problem 19E Let R be a ring. The center of R is the set {x ? R | ax = xa for all a in R}. Prove that the center of a ring is a subring.

Problem 20E Describe the elements of M2(Z) (see Example 4) that have multiplicative inverses. Reference:

Let R be a commutative ring with unity and let U(R) denote the set of units of R. Prove that U(R) is a group under the multiplication of R. (This group is called the group of units of R.)

Problem 23E Determine U(Z[i]) (see Example 11). Reference:

Problem 18E Let a belong to a ring R. Let S = {x ? R | ax = 0}. Show that S is a subring of R.

Problem 21E Suppose that R1, R2, . . . , Rn are rings that contain nonzero elements. Show that R1 ? R2 ? ··· ? Rn has a unity if and only if each Ri has a unity.

If \(R_1, R_2, \ldots , R_n\) are commutative rings with unity, show that \(U\left(R_{1} \oplus R_{2} \oplus \cdots \oplus R_{n}\right)=U\left(R_{1}\right) \oplus U\left(R_{2}\right) \oplus \cdots \oplus U\left(R_{n}\right)\).

Determine U(Z[x]). (This exercise is referred to in Chapter 17.)

Determine U(R[x]).

Problem 27E Show that a unit of a ring divides every element of the ring.

Problem 28E In Z6, show that 4 | 2; in Z8, show that 3 | 7; in Z15, show that 9 | 12

Problem 29E Suppose that a and b belong to a commutative ring R with unity. If a is a unit of R and b2 = 0, show that a + b is a unit of R.

Problem 30E Suppose that there is an integer n > 1 such that xn = x for all elements x of some ring. If m is a positive integer and am = 0 for some a, show that a = 0.

Problem 31E Give an example of ring elements a and b with the properties that ab = 0 but ba ? 0.

Problem 32E Let n be an integer greater than 1. In a ring in which xn = x for all x, show that ab = 0 implies ba = 0.

Suppose that R is a ring such that \(x^3 = x\) for all x in R. Prove that 6x = 0 for all x in R.

Problem 34E Suppose that a belongs to a ring and a4 = a2. Prove that a2n = a2 for all n ? 1.

Problem 35E Find an integer n > 1 such that an = a for all a in Z6. Do the same for Z10. Show that no such n exists for Zm when m is divisible by the square of some prime.

Problem 36E Let m and n be positive integers and let k be the least common multiple of m and n. Show that mZ ? nZ = kZ.

Problem 37E Explain why every subgroup of Zn under addition is also a subring of Zn.

Problem 38E Is Z6 a subring of Z12?

Problem 39E Suppose that R is a ring with unity 1 and a is an element of R such that a2 = 1. Let S = {ara | r ? R}. Prove that S is a subring of R. Does S contain 1?

Problem 40E Let M2(Z) be the ring of all 2 × 2 matrices over the integers and let R = Prove or disprove that R is a subring of M2(Z).

Problem 41E Let M2(Z) be the ring of all 2 × 2 matrices over the integers and let R = Prove or disprove that R is a subring of M2(Z)

Problem 42E Let Prove or disprove that R is a subring of M2(Z)

Problem 44E Suppose that there is a positive even integer n such that an = a for all elements a of some ring. Show that –a = a for all a in the ring.

Problem 43E Let R = Z ? Z ? Z and S = {(a, b, c) ? R | a + b = c}. Prove or disprove that S is a subring of R.

Problem 46E Show that 2Z ? 3Z is not a subring of Z.

Problem 45E Let R be a ring with unity 1. Show that S= {n · 1 | n ? Z} is a subring of R.

Problem 47E Determine the smallest subring of Q that contains 1/2. (That is, find the subring S with the property that S contains 1/2 and, if T is any subring containing 1/2, then T contains S.)

Problem 48E Determine the smallest subring of Q that contains 2/3.

Problem 49E Let R be a ring. Prove that a2 – b2 = (a + b)(a – b) for all a, b in R if and only if R is commutative.

Suppose that R is a ring and that \(a^2 = a\) for all a in R. Show that R is commutative. [A ring in which \(a^2 = a\) for all a is called a Boolean ring, in honor of the English mathematician George Boole (1815–1864).]

Problem 51E Give an example of a Boolean ring with four elements. Give an example of an infinite Boolean ring.