EXERCISES. a b = a + b l aq b = ab - (a + b) + 2. a b = a + b + 1 n0i) = oii + ii + fi. A. Examples of Rings. C. Ring of 2 x 2 Matrices

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1 / rings definitions and elementary properties 171 EXERCISES A. Examples of Rings In each of the following, a set A with operations of addition and multiplication is given. Prove that A satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative ofan arbitrary a. 1 A is the set Z of the integers, with the following "addition" and "multiplication" O a b = a + b l aq b = ab - (a + b) A is the set Q of the rational numbers, and the operations are and O defined as follows a b = a + b + 1 n0i) = oii + ii + fi 3 A is the set Q x Q of ordered pairs of rational numbers, and the operations are the following addition and multiplication (a, = (a + c,b + d) (a, b) Q(c,d) = (ac - bd, ad + be) 4 A = {x + y^jl x, y ez} with conventional addition and multiplication. 5 Prove that the ring in part 1 is an integral domain. 6 Prove that the ring in part 2 is a field, and indicate the multiplicative inverse of an arbitrary nonzero element. 7 Do the same for the ring in part 3. B. Ring of Real Functions 1 Verify that ^"(IR) satisfies all the axioms for being a commutative ring with unity. Indicate the zero and unity, and describe the negative of any 2 Describe the divisors of zero in ^(U). 3 Describe the invertible elements in ^"(IR). 4 Explain why J 5" (R) is neither a field nor an integral domain. C. Ring of 2 x 2 Matrices Let Jt 2 (K) designate the set of all 2 x 2 matrices whose entries are real numbers a, b, c, and d, with the following addition and

2 172 CHAPTER SEVENTEEN multiplication and a b\ fr s\ fa + rb c d) \t u)~\c + t d a b\ fr s\ far + bt as c d)\t u) \cr + dt cs + du) 1 Verify that JK 2 (U) satisfies the ring axioms. 2 Show that J? 2 (R) is commutative and has a unity. 3 Explain why Jt 2 (U) is not an integral domain or a field. D. Rings of Subsets of a Set If D is a set, then the power set of D is the set P D of all the subsets of D. Addition and multiplication are defined as follows If A and B are elements of P D (that is, subsets of D), then A + B = (A - B) u (B - A) and AB = A n B It was shown in Chapter 3, Exercise C, that P D with addition alone is an abelian group. Now prove thefollowing 1 P D is a commutative ring with unity. (You may assume n is associative; for the distributive law, use the same diagram and approach as was used to prove that addition is associative in Chapter 3, Exercise C.) 2 Describe the divisors of zero in P D. 3 Describe the invertible elements in P D. 4 Explain why P D is neither a field nor an integral domain. 5 Give the tables of P 3, that is, P D where D = {1, 2, 3}. E. Ring of Quaternions A quaternion (in matrix form) is a 2 x 2 matrix of complex numbers of the form _ / a + bi c + di\ \ c + di a bi) 1 Prove that the set of all the quaternions, with the matrix addition and multiplication explained on page 8, is a ring with unity. This ring is denoted by the symbol St. Find an example to show that 2. is not commutative. (You may assume matrix addition and multiplication are associative and obey the distributive law.) 2 Let

3 rings definitions and elementary properties 173 Show that the quaternion a, defined previously, may be written in the form a = al + bi + cj + dk (This is the standard notation for quaternions.) 3 Prove the following formulas i 2 = j 2 = k 2 = 1 ij = ji = k jk = kj = i ki = ik = j 4 The conjugate of a is (a bi c di\ c di a + bi) The norm of a is a 2 + b 2 + c 2 + d 2, and is written a. Show directly (by matrix multiplication) that oca = aa = I ft 0\ ) where t = a Conclude that the multiplicative inverse of a is (l/t)a- 5 A skew field is a (not necessarily commutative) ring with unity in which every nonzero element has a multiplicative inverse. Conclude from parts 1 and 4 that Q is a skew field. F. Ring of -Endomorphisms Let G be an abelian group in additive notation. An endomorphism of G is a homomorphism from G to G. Let End(G) denote the set of all the endomorphisms of G, and define addition and multiplication of endomorphisms as follows lf+9~}(x)=f(x ) + g(x) for every x in G Uo\ = f g tne composite of/ and g 1 Prove that End(G) with these operations is a ring with unity. 2 List the elements of End(Z 4 ), then give the addition and multiplication tables for End(Z 4 ). Remark The endomorphisms of Z 4 are easy to find. Any endomorphisms of Z 4 will carry 1 to either 0, 1,2, or 3. For example, take the last case if then necessarily 1 + 1^ = ^ = 1 and 0^0 hence / is completely determined by the fact that

4 174 CHAPTER SEVENTEEN G. Direct Product of Rings If A and B are rings, their direct product is a new ring, denoted by A x B, and defined as follows A x B consists of all the ordered pairs (x, y) where x is in A and y is in B. Addition in A x B consists of adding corresponding components (xi, yi) + (x 2, y 2) = (*i +x 2,yi+ y 2) Multiplication in A x B consists of multiplying corresponding components (*i> yi)-(x 2,y2 ) = (xix 2,y i y 2 ) 1 If A and B are rings, verify that A x B is a ring. 2 If A and B are commutative, show that A x B is commutative. If A and B each has a unity, show that A x B has a unity. 3 Describe carefully the divisors of zero in A x B. 4 Describe the invertible elements in A x B. 5 Explain why A x B can never be an integral domain or a field. (Assume A x B has more than one element.) H. Elementary Properties of Rings Prove each of thefollowing 1 In any ring, a(b c) = ab ac and (b c)a ^ ba ca. 2 In any ring, if ab = ba, then (a + ft) 2 = (a b) 2 = a 2 + b 2. 3 In any integral domain, if a 2 = b 2, then a = ± b. 4 In any integral domain, only 1 and 1 are their own multiplicative inverses. (Note that x = x" 1 iff x 2 = 1.) 5 Show that the commutative law for addition need not be assumed in defining a ring with unity it may be proved from the other axioms. [Hint Use the distributive law to expand (a + b)(l + 1) in two different ways.] 6 Let A be any ring. Prove that if the additive group of A is cyclic, then A is a commutative ring. 7 In any integral domain, if a" = 0 for some integer n, then a = 0. I. Properties of Invertible Elements Prove that each of thefollowing is true in a nontrivial ring with unity. 1 If a is invertible and ab = ac, then b = c. 2 An element a can have no more than one multiplicative inverse. 3 If a 2 = 0 then a + 1 and a 1 are invertible. 4 If a and b are invertible, their product ab is invertible.

5 rings definitions and elementary properties The set S of all the invertible elements in a ring is a multiplicative group. 6 By part 5, the set of all the nonzero elements in a field is a multiplicative group. Now use Lagrange's theorem to prove that in a finite field with m elements, x" 1-1 = 1 for every x # 0. 7 If ax = 1, x is a right inverse of a; if ya = 1, y is a te/f inverse of a. Prove that if a has a right inverse x aurf a left inverse then a is invertible, and its inverse is equal to x and to y. (First show that yaxa =1.) 8 In a commutative ring, if ab is invertible, then a and ft are both invertible. J. Properties of Divisors of Zero Prove that each of thefollowing is true in a nontrivial ring. 1 If a + 1 and a 2 = 1, then a + 1 and a 1 are divisors of zero. 2 If aft is a divisor of zero, then a or ft is a divisor of zero. 3 In a commutative ring with unity, a divisor of zero cannot be invertible. 4 Suppose ab 0 in a commutative ring. If either a or ft is a divisor of zero, so is ab. 5 Suppose a is neither 0 nor a divisor of zero. If ab = ac, then ft = c. 6 A x B always has divisors of zero. K. Boolean Rings A ring A is a boolean ring if a 2 = a for every as A. Prove that each of the following is true in an arbitrary boolean ring A. 1 For every a e A, a = a. [Hint Expand (a + a) 2.] 2 Use part 1 to prove that A is a commutative ring. [Hint Expand (a + ft) 2.] In parts 3 and 4, assume A has a unity. 3 Every element except 0 and 1 is a divisor of zero. [Consider x(x 1).] 4 1 is the only invertible element in A. 5 Letting a V ft = a + ft + aft, we have the following in A flv be = (av ftxav c) a V (1 + a) = 1 ava = a a(avft) = a L. The Binomial Formula An important formula in elementary algebra is the binomial expansion formula for an expression (a + ft)". The formula is as follows

6 176 CHAPTER SEVENTEEN where the binomial coefficient /n\ tin - l)(n - 2) (n - k + 1) W ~ k\ This theorem is true in every commutative ring. (If k is any positive integer and a is an element of a ring, ka refers to the sum a + a + + a with k terms, as in elementary algebra.) The proof of the binomial theorem in a commutative ring is no different from the proof in elementary algebra. We shall review it here. The proof of the binomial formula is by induction on the exponent n. The formula is trivially true for n = 1. In the induction step, we assume the expansion for (a + by is as above, and we must prove that Now, (a + bf +1 = (a + bf +1 = (a + b)(a + b) n n+l ( )a n + l ' k b k fc = 0 V * / = (a + b)i(^a-v = lot>" +, " V + log>"" V+1 Collecting terms, we find that the coefficient of a n+ ~ 1 k b k is (K-,) By direct computation, show that CH-Hr) 1 It will follow that (a + is as claimed, and the proof is complete. M. Nilpotent and Unipotent Elements An element a of a ring is nilpotent if a" = 0 for some positive integer n. 1 In a ring with unity, prove that if a is nilpotent, then a + 1 and a 1 are both invertible. [Hint Use the factorization 1 - a" = (1 +a + a a"~ 1 ) for 1 a, and a similar formula for 1 4- a.] 2 In a commutative ring, prove that any product xa of a nilpotent element a by any element x is nilpotent.

7 rings definitions and elementary properties In a commutative ring, prove that the sum of two nilpotent elements is nilpotent. (Hint You must use the binomial formula; see Exercise L.) An element a of a ring is unipotent iff 1 a is nilpotent. 4 In a commutative ring, prove that the product of two unipotent elements a and b is unipotent. [Hint Use the binomial formula to expand 1 ab = (1 + a) + a(l b) to power h + m.] 5 In a commutative ring, prove that every unipotent element is invertible. (Hint Use the binomial expansion formula.)