# 04/11/2019: Abstract Algebra (I. N. Herstein) /home/zack/Dropbox/Library/I. N. Herstein/Abstract Algebra (405)/Abstract Algebra - I. N. Herstein.pdf Last Annotation: 04/11/2019 ## Highlights - a ring if (I. N. Herstein 142) - \(a\) \(b\) \(c\) \(d\) \(e\) (I. N. Herstein 142) - \(f\) \(g\) (I. N. Herstein 142) - \(h\) (I. N. Herstein 142) - a \. b = 0 we concluded that a = 0 or b = O\. (I. N. Herstein 143) - \. When it does hold, (I. N. Herstein 143) - name; it is called a domain\. (I. N. Herstein 143) - is an integral domain if a \. b = 0 in (I. N. Herstein 143) - R implies that a = 0 or b = O\. (I. N. Herstein 143) - e a division ring if for every (I. N. Herstein 143) - Definition\. A ring R with unit is sa a =I=0 in R there is an element b E R \( (I. N. Herstein 143) - s a-I\) (I. N. Herstein 143) - a \. a-I = a-I\. a = 1\. (I. N. Herstein 143) - A ring R is said to be afield if R is a commutative division (I. N. Herstein 143) - rlng\. (I. N. Herstein 143) - , R is a field if the nonzero ele- (I. N. Herstein 143) - ments of R form an abelian group under· , the product in R\. (I. N. Herstein 143) - e examples (I. N. Herstein 144) - of rings\. W (I. N. Herstein 144) - y 7L, (I. N. Herstein 144) - , 7L is an example of an integral domain\. (I. N. Herstein 144) - o Q is a field\. (I. N. Herstein 144) - s, IR, also give us an example of a field\. (I. N. Herstein 144) - s, C, form a field\. (I. N. Herstein 144) - t R = 7L 6 , (I. N. Herstein 144) - 7L 6 is not an integral domain, (I. N. Herstein 144) - \. R is a commutative ring with unit\. (I. N. Herstein 144) - t a =1= 0 in a ring R is a zero-divisor in R if ab =0 (I. N. Herstein 144) - for some b =1= 0 in R\. (I. N. Herstein 144) - n 7L p is clearly a (I. N. Herstein 145) - commutative ring with 1\. (I. N. Herstein 145) - t 7L p is a field\. T (I. N. Herstein 145) - 7L p has only a finite number of elements, it is called a finite (I. N. Herstein 145) - field\. (I. N. Herstein 145) - et R be (I. N. Herstein 145) - the set of all a E Q in whose reduced form the denominator is odd\. (I. N. Herstein 145) - t R forms a ring\. It is an inte- (I. N. Herstein 145) - gral domain with unit but is not a field, f (I. N. Herstein 145) - l a E Q in whose reduced form the denominator is (I. N. Herstein 145) - not divisible by a fixed prime p\. (I. N. Herstein 145) - , is an integral domain but is not a field\. (I. N. Herstein 145) - \), R is a ring u (I. N. Herstein 145) - t R be the set of all real-valued continuous functions on the closed unit (I. N. Herstein 145) - interval [0, 1]\. F (I. N. Herstein 145) - e \(f + g\)\(x\) = f \(x\) + g \(x\), (I. N. Herstein 145) - \(f· g\)\(x\) = f\(x\)g \(x\)\. F (I. N. Herstein 145) - s R is a commuta- (I. N. Herstein 145) - tive rin (I. N. Herstein 145) - g\. It is not an integral domain\. F (I. N. Herstein 145) - For S to be a subring, it is necessary and sufficient that S be nonempty (I. N. Herstein 145) - and that ab, a + b E S for all a, b E S\. (I. N. Herstein 145) - a subring of R is a subset S of R which (I. N. Herstein 145) - is a ring if the operations ab and a + b are just the operations of R applied to (I. N. Herstein 145) - the elements a, b E S\. (I. N. Herstein 145) - n i 2 = j2 = (I. N. Herstein 148) - k 2 = -1,ij=k,jk=i,ki=jan (I. N. Herstein 148) - dji= -k,kj= -i,ik= -j\.If (I. N. Herstein 148) - i i @ S 2\. S (I. N. Herstein 149) - s the s ¢ quaternions guaiernions (I. N. Herstein 149) - o form POO N JLIETTL o a& noncommutative A \ o Selifelasiaa N it e VO division RORCOMMEIaiive A givision ring\. QR W @A W VL singe -y FeiiX\. S s s AN TR Wt LAY oy ¢ o L= (I. N. Herstein 149) - @F 8\. F (I. N. Herstein 150) - 10\. (I. N. Herstein 150) - 14\. S 15\. F (I. N. Herstein 151) - 14\. S 15\. F 16\. V (I. N. Herstein 151) - 19\. S q (I. N. Herstein 151) - q 20\. I (I. N. Herstein 151) - 21\. S 22\. i (I. N. Herstein 151) - 33\. L d (I. N. Herstein 152) - ety 36\. I 37\. I (I. N. Herstein 153) - 39\. I 39\. I \(40, P 40\. P (I. N. Herstein 153) - n integral domain (I. N. Herstein 155) - \) cp\(a + b\) = cp \(a\) + cp\(b\) a (I. N. Herstein 155) - cp\(ab\) = cp\(a\)cp\(b\) f (I. N. Herstein 155) - , the image of R under (I. N. Herstein 156) - a homomorphism from R to R', is a subring of R', (I. N. Herstein 156) - Ker cp is an additive subgroup of R\. (I. N. Herstein 156) - \. So Ker cp swallows up multiplication (I. N. Herstein 156) - from the left and the right by arbitrary ring elements\. (I. N. Herstein 156) - ideal of R if: (I. N. Herstein 156) - \(a\) I \(b\) G (I. N. Herstein 156) - Ker cp is an (I. N. Herstein 156) - ideal of R\. (I. N. Herstein 156) - t every ideal can be made the kernel of a homo- (I. N. Herstein 156) - morphism\. (I. N. Herstein 156) - t well-defined\. (I. N. Herstein 157) - e, the (I. N. Herstein 157) - So R/K is now endowed with a sum and a product\. Furthermore, the mapping cp: R ~ R/K defined by cp\(a\) = a + K for a E R is a homomor- (I. N. Herstein 157) - phism of R onto R/K with kernel K\. \( (I. N. Herstein 157) - R/K is a ring, (I. N. Herstein 157) - So R/K is a homo- (I. N. Herstein 157) - morphic image of R\. (I. N. Herstein 157) - n R' ~ RIK; (I. N. Herstein 158) - 'P: R ~ R' b (I. N. Herstein 158) - tfJ: RIK ~ R' (I. N. Herstein 158) - y tfJ\(a + K\) = 'P\(a\) (I. N. Herstein 158) - isomorphism of RIK onto R'\. (I. N. Herstein 158) - f I' is an ideal of R', (I. N. Herstein 158) - I = {a E R I 'P\(a\) E I'}\. Then I is an ideal of R, I => K and 11K ~ I'\. T (I. N. Herstein 158) - a 1-1 correspondence between all the ideals of R' and those ideals of R (I. N. Herstein 158) - that contain K\. (I. N. Herstein 158) - n A + I = { (I. N. Herstein 158) - subring of R, I is an ideal of A + I, and \(A + 1\)1I ~ A I\(A n I\)\. (I. N. Herstein 158) - n RII ~ R'II'\. (I. N. Herstein 158) - d I => K is (I. N. Herstein 158) - n RII ~ \(RIK\)/\(IIK\)\. (I. N. Herstein 158) - 7L, the ring of integers, f (I. N. Herstein 158) - t In be the set of all multiples of n; t (I. N. Herstein 158) - t R be the ring of all rational numbers having odd denominators in their (I. N. Herstein 158) - reduced form (I. N. Herstein 158) - t I b (I. N. Herstein 158) - even numerator; (I. N. Herstein 159) - e cp : R ~» : ~ 7\.2 , 7L \_ Ay ! < (I. N. Herstein 159) - v ' y@{a’b\) == 00 if cp\(a/b\) iy Y if aa is g d is even even \(a {a (I. N. Herstein 159) - efalb\)y cp \(a/b\) == 11 if a¢ is 15 odd\.W / W (I. N. Herstein 159) - &, 2 ===== &/{\. s 7L R/I\. G (I. N. Herstein 159) - t & bethe RIy be E ey e oy sud e 1T e st ot T oh T 1 et T ey vy Nur I the ring of all rational numbers whose denominators ey (I. N. Herstein 159) - ivisibleby p, are not divisible p, (I. N. Herstein 159) - t I§§ bebe those t el those elements in (I. N. Herstein 159) - & nuimerator is R whose numerator Pi e TR Y S e s divisible by p; ¥ L p; I (I. N. Herstein 159) - t {I == {f E t &€ Rif{&\) R If\(~\) IJ \\.\.\. = OJ\. = \(]\. (I. N. Herstein 159) - T % 'L\.\.q o & i « < 9\. \. § o \.° § m \. ¢ \_ N o < 5 \. t R t FANE be VIR the B Ll ring 3§ 5524 of all Al real-valued FEA-Vailugcyd continuous CONTIMIONUS functions FICTIONS on O the 1830 closed Ci08Sa unit 7Oy TS TNy \ N e al xPAITISO OSOantien e\\ 3-’\ TIQ M :\} ;\ O io e"§ I t | w1y (I. N. Herstein 159) - interval w interval w (I. N. Herstein 159) - What is 'lll/ 1s R/I? 8/{?7 (I. N. Herstein 159) - cp\(f\) «{f\) == f\(~\)\.f{i\)\. (I. N. Herstein 159) - t cp: ¢: &R— ~ & IR (I. N. Herstein 159) - N 33 ~ ~\) @yL a, bb E& IR} ; ¥ N RR {'g {3,} E 5 i~ 31\ e ?“\‘\4‘ é\ \. t R = {\( 33 R VU sy 3§ 5 : ls\. (I. N. Herstein 160) - &g 3 i Y \(~ ~\)\.1 b&R}li 8 eN v Xi Vi 5 y & RV eti=4{{, et I ={ R & 83 v b E IR} I i i N CVU ideal of R? Consider (I. N. Herstein 160) - at R/I at &/{ "'" == IR ¥ aa FRPT R (I. N. Herstein 160) - at I == Ker cpo So R/I = image of cp == IR (I. N. Herstein 161) - et R = {\( \_: ~\) a, b E IR} an (I. N. Herstein 161) - by l/J \( \_: ~ \) = a + bi\. W sm of R onto C\. So R is (I. N. Herstein 161) - t R be any commutative ring with 1\. I (I. N. Herstein 161) - t \(a\) is an ideal of R\. T (I. N. Herstein 161) - Let R be a commutative ring with unit whose only (I. N. Herstein 164) - Lemma 4\.4\.1\. Let R be a commu ideals are \(0\) and itself\. Then R is a field\. (I. N. Herstein 164) - A proper ideal M of R is a maximal ideal of R if the only (I. N. Herstein 164) - ideals of R that contain Mare M itself and R\. (I. N. Herstein 164) - t M be a (I. N. Herstein 165) - maximal ideal of R\. Then RIM is a field\. (I. N. Herstein 165) - t R be a commutative ring with 1, (I. N. Herstein 165) - the greatest common divisor always be a (I. N. Herstein 173) - monic polynomial\. (I. N. Herstein 173) - The polynomial p \(x\) E F[x] is irreducible if p \(x\) is of pos- (I. N. Herstein 175) - itive degree and given any polynomial f\(x\) in F[x], then either p\(x\) I f\(x\) or (I. N. Herstein 175) - p\(x\) is relatively prime to f\(x\)\. (I. N. Herstein 175) - If p\(x\) E F[x] , then the ideal \(p\(x» generated by (I. N. Herstein 176) - p\(x\) in F[x] is a maximal ideal of F[x] if and only if p\(x\) is irreducible in F[x]\. (I. N. Herstein 176) - e it tells us exactly what the maximal (I. N. Herstein 177) - ideals of F[x] are, namely the ideals generated by the irreducible polynomi- (I. N. Herstein 177) - als\. (I. N. Herstein 177) - \. If M is a maximal ideal of F[x], F[x]/M is a field, and this field contains (I. N. Herstein 177) - F \( (I. N. Herstein 177) - An integral domain R is a Euclidean ring if there is a (I. N. Herstein 179) - function d from the nonzero elements of R to the nonnegative integers that (I. N. Herstein 179) - satisfies: (I. N. Herstein 179) - \(a\) \(b\) (I. N. Herstein 179)