464 ALGEBRAIC SYSTEMS [APP. B
RINGS
B.69. Consider the ringZ 12 ={ 0 , 1 ,..., 11 }of integers modulo 12. (a) Find the units ofZ 12. (b) Find the roots of
f(x)=x^2 + 4 x+4 overZ 12. (c) Find the associates of 2.
B.70. Consider the ringZ 30 ={ 0 , 1 ,..., 29 }of integers modulo 30.
(a) Find−2,−7, and−11. (b) Find: 7−^1 ,11−^1 , and 26−^1.
B.71. Show that in a ringR: (a) (−a)(−b)=ab; (b)(− 1 )(− 1 )=1, ifRhas an identity element 1.
B.72. Supposea^2 =afor everya∈R. (Such a ring is called aBooleanring). Prove thatRis commutative.
B.73. LetRbe a ring with an identity element 1. We makeRinto another ringR′by defining:a⊕b=a+b+1 and a∗b=ab+a+b(a) Verify thatR′is a ring. (b) Determine the 0-element and the 1-element ofR′.
B.74. LetGbe any (additive) abelian group. Define a multiplication inGbya∗b=0 for everya, b∈G. Show that this
makesGinto a ring.
B.75. LetJandKbe ideals in a ringR. Prove thatJ+KndJ∩Kare also ideals.
B.76. LetRbe a ring with unity 1. Show that(a)={ra|r∈R}is the smallest ideal containinga.
B.77. Show thatRand {0} are ideals of any ringR.
B.78. Prove: (a) The units of a ringRform a group under multiplication. (b) The units inZmare those integers which are
relatively prime tom.
B.79. For any positive integerm, verify thatmZ={rm|r∈Z}is a ring. Show that 2Zand 3Zare not isomorphic.
B.80. Prove Theorem B.10: LetJbe an ideal in a ringR. Then the cosets{a+J|a ∈R}form a ring under the coset
operations (a+J)+(b+J)=a+b+Jand (a+J )(b+J)=ab+J.
B.81. Prove Theorem B.11: Letf:R→R′be a ring homomorphism with kernelK. ThenKis an ideal inR, and the quotient
ringR/Kis isomorphic tof(R).
B.82. LetJbe an ideal in a ringR. Consider the (canonical) mappingf:R→R/Jdefined byf(a)=a+J. Show that:
(a)fis a ring homomorphism; (b)fis an onto mapping.
B.83. SupposeJis an ideal in a ringR. Show that: (a) IfRis commutative, thenR/Jis commutative. (b) IfRhas a unity
element 1 and 1∈J, then 1+Ris a unity element forR/J.INTEGRAL DOMAINS AND FIELDS
B.84. Prove that ifx^2 =1 in an integral domainD, thenx=−1orx=1.
B.85. LetR={ 0 }be a finite commutative ring with no zero divisors. Show thatRis an integral domain, that is, thatRhas
an identity element 1.
B.86. Prove thatF={a+b√
2 |a, brational}is a field.
B.87. Prove thatF={a+b√
2 |a, bintegers}is an integral domain but not a field.
B.88. Acomplex numbera+biwhere a,bare integers is called aGaussian integer. Show that the setGof Gaussian integers
is an integral domain. Also show that the units are±1,±i.
B.89. LetRbe an integral domain and letJbe an ideal inR. Prove that the factor ringR/Jis an integral domain if and only
ifJis a prime ideal. (An idealJisprimeifJ=Rand ifab∈Jimpliesa∈Jorb∈J.)
B.90. LetRbe a commutative ring with unity element 1, and letJbe an ideal inR. Prove that the factor ringR/Jis a field
if and only ifJis a maximal ideal. (An idealJis maximal ifJ=Rand no idealKlies strictly betweenJandR, that
is, ifJ⊆K⊆RthenJ=KorK=R.)B.91. LetDbethe ring of real 2× 2 matrices of the form[
a −b
ba]. Show thatDis isomorphic to the complex fieldC,
whenDis a field.
B.92. Show that the only ideal in a fieldKis {0} orKitself.
B.93. Supposef: K→K′is a homomorphism from a fieldKto a fieldK′. Show thatfis anembedding; that is,fis
one-to-one. (We assumef(1)=0.)