Because we have quotient rings and ring homomorphisms, we actually get a number of “theorems” for free. These are the isomorphism theorems for rings, the most important one being the analogue of the First Isomorphism Theorem for groups. That is, if f:R→S is a ring homomorphism, im(f...
Basic Algebr a II Second Editio n N A T H A N J A C O B S O N YAL E UNIVERSIT Y • B W . H . F R E E M A N A N D C O M P A N Y N ew Yor k To Mik e and Polly Library of Congress Cataloging-in-Publication Data (Revised for vol. 2) Jacobson, Nathan, ...
Proof of Theorem 5. We have seen that if 𝑿⊗𝒀X⊗Y is regular of dimension 1, then 𝝀λ is an isomorphism. On the other hand, assume that 𝝀λ is surjective. Then, by the exact sequence in Proposition 6, we see that ∧𝟐(Ω𝟏(𝑿⊗𝒀))=𝟎.∧2(Ω1(X⊗Y...