Suppose that a and b belong to a commutative ring and ab

Problem 2CE Exercise 1. This software lists all idempotents (see the chapter exercises for the definition) in Z n . Run the program for various values of n . Use these data to make conjectures about the number of idempotents in Z n as a function of n. For example, how many idempotents are there when n is a prime power? What about when n is divisible by exactly two distinct primes? In the case where n is of the form pq where p and q are primes can you see a relationship between the two idempotents that are not 0 and 1? Can you see a relationship between the number of idempotents for a given n and the number of distinct prime divisors of n ? Exercise 2. This software lists all nilpotent elements (see the chapter exercises for definition) in Z n . Run your program for various values of n . Use these data to make conjectures about nilpotent elements in Z n as a function of n . Exercise 3. This software determines which rings of the form Z p [ i ] are fields. Run the program for all primes up to 37. From these data, make a conjecture about the form of the primes that yield a field. Exercise 1. This software lists all idempotents (see the chapter exercises for the definition) in Z n . Run the program for various values of n . Use these data to make conjectures about the number of idempotents in Z n as a function of n. For example, how many idempotents are there when n is a prime power? What about when n is divisible by exactly two distinct primes? In the case where n is of the form pq where p and q are primes can you see a relationship between the two idempotents that are not 0 and 1? Can you see a relationship between the number of idempotents for a given n and the number of distinct prime divisors of n ? Exercise 2. This software lists all nilpotent elements (see the chapter exercises for definition) in Z n . Run your program for various values of n . Use these data to make conjectures about nilpotent elements in Z n as a function of n . Exercise 3. This software determines which rings of the form Z p [ i ] are fields. Run the program for all primes up to 37. From these data, make a conjecture about the form of the primes that yield a field.

Problem 1CE Exercise 1. This software lists all idempotents (see the chapter exercises for the definition) in Z n . Run the program for various values of n . Use these data to make conjectures about the number of idempotents in Z n as a function of n. For example, how many idempotents are there when n is a prime power? What about when n is divisible by exactly two distinct primes? In the case where n is of the form pq where p and q are primes can you see a relationship between the two idempotents that are not 0 and 1? Can you see a relationship between the number of idempotents for a given n and the number of distinct prime divisors of n ? Exercise 2. This software lists all nilpotent elements (see the chapter exercises for definition) in Z n . Run your program for various values of n . Use these data to make conjectures about nilpotent elements in Z n as a function of n . Exercise 3. This software determines which rings of the form Z p [ i ] are fields. Run the program for all primes up to 37. From these data, make a conjecture about the form of the primes that yield a field.

Problem 3CE Exercise 1. This software lists all idempotents (see the chapter exercises for the definition) in Z n . Run the program for various values of n . Use these data to make conjectures about the number of idempotents in Z n as a function of n. For example, how many idempotents are there when n is a prime power? What about when n is divisible by exactly two distinct primes? In the case where n is of the form pq where p and q are primes can you see a relationship between the two idempotents that are not 0 and 1? Can you see a relationship between the number of idempotents for a given n and the number of distinct prime divisors of n ? Exercise 2. This software lists all nilpotent elements (see the chapter exercises for definition) in Z n . Run your program for various values of n . Use these data to make conjectures about nilpotent elements in Z n as a function of n . Exercise 3. This software determines which rings of the form Z p [ i ] are fields. Run the program for all primes up to 37. From these data, make a conjecture about the form of the primes that yield a field.

Problem 70E Let with the usual matrix addition and multiplication and mod 7 addition and multiplication of the entries. Prove that R is a commutative ring. How many elements are in R? Is R a field? What happens when Z7 is replaced by Z5?

Which of Examples 1 through 5 are fields?

Verify that Examples 1 through 8 are as claimed. Reference: EXAMPLE 1 The ring of integers is an integral domain EXAMPLE 2 The ring of Gaussian integers \(Z[i]=\{a+b i \mid a, b \in Z\}\) is an integral domain. EXAMPLE 3 The ring \(Z[x]\) of polynomials with integer coefficients is an integral domain. EXAMPLE 4 The ring \(Z[\sqrt{2}]=\{a+b \sqrt{2} \mid a, b \in Z\}\) is an integral domain. EXAMPLE 5 The ring \(Zp\) of integers modulo a prime \(p\) is an integral domain. EXAMPLE 6 The ring \(Zn\) of integers modulo \(n\) is not an integral domain when \(n\) is not prime. EXAMPLE 7 The ring \(M 2(Z) \text { of } 2 \times 2\) matrices over the integers is not an integral domain. EXAMPLE 8 \(Z \oplus Z\) is not an integral domain.

Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.

Problem 4CE This software finds the idempotents in Zn[i] = {a+bi| a, b belong to Zn} (Gaussian integers modulo n ). Run the software for n = 4, 8, 16, and 32. Make a conjecture about the number of idempotents when n = 2k. Run the software for n = 13, 17, 29, and 37. What do these values of n have in common? Make a conjecture about the number of idempotents for these n. Run the software for n = 7, 11, 19, 23, 31, and 43. What do these values of n have in common? Make a conjecture about the number of idempotents for these n.

Problem 4E List all zero-divisors in Z20. Can you see a relationship between the zero-divisors of Z20 and the units of Z20?

Problem 5CE This software finds the nilpotent elements in Z n [ i ] = { a + bi | a and b both belong to Z n }. Run the software for n = 4, 8, 16, and 32. Make a conjecture about the number of nilpotent elements when n = 2 k . Run the software for n = 3, 5, 7, 11, 13, and 17. What do these values for n have in common? Make a conjecture about the number of nilpotent elements for these n . Run the program for n = 9 . Do you need to revise the conjecture you make based on n = 3, 5, 7, 11, 13 , and 17? Run the software for n = 9, 25, and 49. What do these values for n have in common? Make a conjecture about the number of nilpotent elements for these n . Run the program for n = 27 . Do you need to revise the conjecture you made based on n = 9, 25, and 49? Run your program for n = 125 (this may take a few seconds.) On the basis of all of your data for this exercise make a single conjecture in the case that n = p k where p is any prime. Run the program for n = 6, 15, and 21. Make a conjecture. Run the program for 12, 20, 28, and 45. Make a conjecture. Run the program for 36 and 100 (this may take a few minutes). On the basis of all your data for this exercise make a single conjecture that covers all integers n > 1 .

Problem 5E Show that every nonzero element of Zn is a unit or a zero-divisor.

Problem 6CE This software determines the zero-divisors in Zn [i] = { a+bi | a, b belong to Zn }. Use the software to formulate and test conjectures about the number of zero-divisors in Zn[i] based on various conditions of n. Run the program for n = 5, 13, and 17. Make conjecture about the number of zero-divisors for primes of this form. Run the program for n = 3, 7, 11, and 19. Make conjecture about the number of zero-divisors for primes of this form.

Find a nonzero element in a ring that is neither a zero-divisor nor a unit.

Problem 7E Let R be a finite commutative ring with unity. Prove that every nonzero element of R is either a zero-divisor or a unit. What happens if we drop the “finite” condition on R?

Let \(a \ne 0\) belong to a commutative ring. Prove that a is a zero divisor if and only if \(a^2b = 0\) for some \(b \ne 0 \).

Find elements a, b, and c in the ring \(Z \oplus Z \oplus Z\) such that ab, ac, and bc are zero-divisors but abc is not a zero-divisor.

Problem 10E Describe all zero-divisors and units of Z ? Q ? Z.

Let d be an integer. Prove that \(Z[\sqrt d] = \{a + b \sqrt d | a, b \in Z\}\) is an integral domain. (This exercise is referred to in Chapter 18.)

Problem 12E In Z7, give a reasonable interpretation for the expressions 1/2, .

Problem 13E Give an example of a commutative ring without zero-divisors that is not an integral domain.

Problem 14E Find two elements a and b in a ring such that both a and b are zerodivisors, a + b ? 0, and a + b is not a zero-divisor.

Problem 15E Let a belong to a ring R with unity and suppose that an = 0 for some positive integer n. (Such an element is called nilpotent.) Prove that 1 – a has a multiplicative inverse in R. [Hint: Consider (1 – a)(1 + a + a2 + ··· + an–1).]

Problem 16E Show that the nilpotent elements of a commutative ring form a subring.

Problem 17E Show that 0 is the only nilpotent element in an integral domain.

Problem 18E A ring element a is called an idempotent if a2 = a. Prove that the only idempotents in an integral domain are 0 and 1.

Problem 19E Let a and b be idempotents in a commutative ring. Show that each of the following is also an idempotent: ab, a – ab, a + b – ab, a + b – 2ab.

Problem 20E Show that Zn has a nonzero nilpotent element if and only if n is divisible by the square of some prime.

Problem 21E Let R be the ring of real-valued continuous functions on [–1, 1]. Show that R has zero-divisors.

Prove that if a is a ring idempotent, then \(a^n =a\) for all positive integers n.

Problem 23E Determine all ring elements that are both nilpotent elements and idempotents.

Find a zero-divisor in \(Z_{5}[i]=\left\{a+b i \mid a, b \in Z_{5}\right\}\).

Problem 25E Find an idempotent in Z5[i] = {a + bi | a, b ? Z5}.

Problem 26E Find all units, zero-divisors, idempotents, and nilpotent elements in Z3 ? Z6.

Problem 27E Determine all elements of a ring that are both units and idempotents.

Problem 28E Let R be the set of all real-valued functions defined for all real numbers under function addition and multiplication. a. Determine all zero-divisors of R. b. Determine all nilpotent elements of R. c. Show that every nonzero element is a zero-divisor or a unit.

Problem 29E (Subfield Test) Let F be a field and let K be a subset of F with at least two elements. Prove that K is a subfield of F if, for any a, b (b ? 0) in K, a – b and ab–1 belong to K.

Problem 30E Let d be a positive integer. Prove that a, b ? Q} is a field.

Let R be a ring with unity 1. If the product of any pair of nonzero elements of R is nonzero, prove that ab = 1 implies ba = 1.

Problem 32E Let R = {0, 2, 4, 6, 8} under addition and multiplication modulo 10. Prove that R is a field.

Formulate the appropriate definition of a subdomain (that is, a "sub" integral domain). Let D be an integral domain with unity 1. Show that \(P=\{n \cdot 1 \mid n \in Z\}\) (that is, all integral multiples of 1 ) is a subdomain of D. Show that P is contained in every subdomain of D. What can we say about the order of P ?

Prove that there is no integral domain with exactly six elements. Can your argument be adapted to show that there is no integral domain with exactly four elements? What about 15 elements? Use these observations to guess a general result about the number of elements in a finite integral domain.

Problem 35E Let F be a field of order 2n. Prove that char F = 2.

Determine all elements of an integral domain that are their own inverses under multiplication.

Problem 37E Characterize those integral domains for which 1 is the only element that is its own multiplicative inverse.

Problem 38E Determine all integers n > 1 for which (n – 1)! is a zero-divisor in Zn.

Suppose that a and b belong to an integral domain. a. If \(a^5=b^5\) and \(a^3=b^3\), prove that a = b. b. If \(a^m=b^m\) and \(a^n=b^n\), where m and n are positive integers that

Problem 40E Find an example of an integral domain and distinct positive integers m and n such that am = bm and an = bn, but a ? b.

Problem 41E If a is an idempotent in a commutative ring, show that 1 – a is also an idempotent.

Problem 42E Construct a multiplication table for Z2[i], the ring of Gaussian integers modulo 2. Is this ring a field? Is it an integral domain?

The nonzero elements of \(Z_3[i]\) form an Abelian group of order 8 under multiplication. Is it isomorphic to \(Z_{8}, Z_{4} \oplus Z_{2}, \text { or } Z_{2} \oplus Z_{2} \oplus Z_{2}\)?

Problem 44E Show that is a field. For any positive integer k and any prime p, determine a necessary and sufficient condition for to be a field.

Problem 45E Show that a finite commutative ring with no zero-divisors and at least two elements has a unity.

Problem 46E Suppose that a and b belong to a commutative ring and ab is a zero- divisor. Show that either a or b is a zero-divisor.

Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.

Problem 48E Suppose that R is a commutative ring without zero-divisors. Show that the characteristic of R is 0 or prime.

Problem 49E Let x and y belong to a commutative ring R with prime characteristic p. a. Show that (x + y) p = xp + yp. b. Show that, for all positive integers n, (x + y) pn = xpn + ypn. c. Find elements x and y in a ring of characteristic 4 such that (x + y)4 ? x4 + y4. (This exercise is referred to in Chapter 20.)

Problem 50E Let R be a commutative ring with unity 1 and prime characteristic. If a ? R is nilpotent, prove that there is a positive integer k such that (1 + a)k = 1.

Problem 51E Show that any finite field has order pn, where p is a prime. Hint: Use facts about finite Abelian groups. (This exercise is referred to in Chapter 22.)

Problem 52E Give an example of an infinite integral domain that has characteristic 3.

Problem 53E Let R be a ring and let M2(R) be the ring of 2 × 2 matrices with entries from R. Explain why these two rings have the same characteristic.

Problem 54E Let R be a ring with m elements. Show that the characteristic of R divides m.

Problem 55E Explain why a finite ring must have a nonzero characteristic.

Problem 56E Find all solutions of x2 – x + 2 = 0 over Z3[i]. (See Example 9.)

Problem 57E Consider the equation x2 – 5x + 6 = 0. a. How many solutions does this equation have in Z7? b. Find all solutions of this equation in Z8. c. Find all solutions of this equation in Z12. d. Find all solutions of this equation in Z14.

Problem 58E Find the characteristic of Z4 ? 4Z.

Suppose that R is an integral domain in which \(20 \cdot 1=0\) and \(12 \cdot 1=0\). (Recall that \(n \cdot 1\) means the sum \(1+1+\cdots+1\) with n terms.) What is the characteristic of R?

In a commutative ring of characteristic 2, prove that the idempotents form a subring.

Problem 61E Describe the smallest subfield of the field of real numbers that contains . (That is, describe the subfield K with the property that K contains and if F is any subfield containing , then F contains K.)

Problem 62E Let F be a finite field with n elements. Prove that xn–1 = 1 for all nonzero x in F.

Let F be a field of prime characteristic p. Prove that \(K=\left\{x \in F \mid x^{p}=x\right\}\) is a subfield of F.

Suppose that a and b belong to a field of order 8 and that \(a^2 + ab + b^2 = 0\). Prove that a = 0 and b = 0. Do the same when the field has order \(2^n\) with n odd.

Problem 65E Let F be a field of characteristic 2 with more than two elements. Show that (x + y)3 ? x3 + y3 for some x and y in F.

Problem 66E Suppose that F is a field with characteristic not 2, and that the nonzero elements of F form a cyclic group under multiplication. Prove that F is finite.

Problem 67E Suppose that D is an integral domain and that ? is a nonconstant function from D to the nonnegative integers such that ?(xy) = ?(x)?(y). If x is a unit in D, show that ?(x) = 1.

Let F be a field of order 32. Show that the only subfields of F are F itself and {0, 1}.

Problem 69E Suppose that F is a field with 27 elements. Show that for every element a ? F, 5a = –a.