By J. P. Levine

ISBN-10: 3540097392

ISBN-13: 9783540097396

ISBN-10: 354038555X

ISBN-13: 9783540385554

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**Extra resources for Algebraic Structure of Knot Modules**

**Example text**

For if they generated 36 a proper K ideal J is a d i r e c t of R, summand. K c J~J then --this is i m p o s s i b l e , since Write: 1 = x~ + yB + oX 0 + ~ 0 The e l e m e n t Therefore ~ = ~0 ~ + ~0 B c ~S ~ Y = ~T + Y N we can c h o o s e . . g e, ~ . . Suppose ~ ~' T 8. (~' P) = g(B' and so ~T, ~ ~' so that check ~S. ~0 )' (ii)k+ 1 Since ~ - B = ~(~ T - ~ o)o = ¢(~ ~ - ~ = ¢(¢ ~ - ~ = (~(~7 + and so ~ e - ~o ~ - ~(XoO ~)~ + n(Xo~ + ~o ~ - ~(1 ~)o + n((~ @ ~, ~) x Now set = (~', B'). Thus - o @) + (-~ ~ + n ~0 ) (~ - T 0) + n(~o~ ~(-~ T' ~, n c R..

N-primary is projective or surjectivity k is also elementary is projective. ~: A ~ B can be lifted is injective choose A any ~-primary if and only if Note that A~B is an R-module A is injective. 4: implies be an elementary ~: A 0 + B 0 a A-module Then is free over which Dedekind exists Lemma 0 ~ A0~M A/(~d), Let ~ B 0 ~ M. = O. the same for If Then elementary. ~ = k~, • ~d-iA + ~d-IB. Since ~ ~(a) k~ ~ Ker ¢. 5: Let A, degree, with a' e A = ~ k+l B ~ B and so . C¢(A) be e l e m e n t a r y Dedekind.

Suppose ~ ~' T 8. (~' P) = g(B' and so ~T, ~ ~' so that check ~S. ~0 )' (ii)k+ 1 Since ~ - B = ~(~ T - ~ o)o = ¢(~ ~ - ~ = ¢(¢ ~ - ~ = (~(~7 + and so ~ e - ~o ~ - ~(XoO ~)~ + n(Xo~ + ~o ~ - ~(1 ~)o + n((~ @ ~, ~) x Now set = (~', B'). Thus - o @) + (-~ ~ + n ~0 ) (~ - T 0) + n(~o~ ~(-~ T' ~, n c R.. • + + for ~ ~ x) B ~ - @ ¢ ~T. (ii) k. We may w r i t e for some - ~ by ~", B" ¢ ~S, + ~ B' = (~ B + q ~ 0 ) ( ~ ~(~ 0 c ~k+ls ~k+is (X, ~) c K. -~) + q(~0' = But We w i l l + p B' c T' A X ~ + ~ ~ ¢ ~S, modulo O c TN ~k+is, @) + + n(~o~ o(x ~0 ~ @ + P0 ~ P0 ~ e) - + ~o~)) - ~ - x + y ~ - ~ ) ) ~)) + n y))e @ B c ~k+lT c ~.

### Algebraic Structure of Knot Modules by J. P. Levine

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